Egerváry Research Group
on Combinatorial Optimization
Technical reportS
TR-2010-13. Published by the Egerváry Research Group, Pázmány P. sétány 1/C,
H–1117, Budapest, Hungary. Web site: www.cs.elte.hu/egres . ISSN 1587–4451.
Tree metrics and edge-disjoint
S-paths
Hiroshi Hirai and Gyula Pap
December 2010
1
EGRES Technical Report No. 2010-13
Tree metrics and edge-disjoint S-paths
Hiroshi Hirai
??
and Gyula Pap
?
???
Abstract
Given an undirected graph G = (V, E) with a terminal set S, a terminal
weight µ : S2 → Z+ , and an edge-cost a : E → Z+ , the µ-weighted
minimumP
cost edge-disjoint S-paths problem (µ-CEDP) is to maximize P ∈P µ(sP , tP )−
a(P ) over all edge-disjoint sets P of S-paths, where sP , tP denote the ends of
P and a(P ) is the sum of edge-cost a(e) over edges e in P .
Our main result is a complete characterization of terminal weights µ for
which µ-CEDP is tractable and admits a combinatorial min-max theorem for every graph. We prove that if µ is a tree metric, then µ-CEDP is solvable in polynomial time and has a combinatorial min-max formula, which extends Mader’s
edge-disjoint S-paths theorem and its minimum-cost version by Karzanov. Our
min-max theorem includes the dual half-integrality, which was earlier conjectured by Karzanov for a special case. We also prove that the cost-less version
(µ-EDP) is NP-hard if µ is not a truncated tree metric, where a truncated tree
metric is a weight function represented as pairwise distances among balls in a
tree. On the other hand, µ-EDP for a truncated tree metric µ reduces to µ0 CEDP for a tree metric µ0 . Thus our result is best possible unless P = N P . As
an application, we get a good approximation algorithm for µ-EDP with “near”
tree metric µ by utilizing results from the theory of low-distortion embedding.
1
Introduction
Let G = (V, E) be an undirected graph with a terminal set S ⊆ V . An S-path is a
path connecting distinct terminals in S. Let a : E → Z+ be a nonnegative integral
edge-cost, and let µ : S2 → Z+ be a nonnegative integral terminal weight defined
on the set S2 of pairs on S. We address the µ-weighted minimum-cost edge-disjoint
?
The first author is supported by a Grant-in-Aid for Scientific Research from the Ministry of
Education, Culture, Sports, Science and Technology of Japan.
??
Department of Mathematical Informatics, University of Tokyo, Tokyo 113-8656, Japan,
[email protected]
???
MTA-ELTE Egerváry Research Group (EGRES). Department of Operations Research,
Eötvös Loránd University, Pázmány Péter sétány, 1/C, 1117 Budapest, Hungary. E-mail:
[email protected]. Supported by the Hungarian National Foundation for Scientific Research
(OTKA) grant CK80124.
December 2010
2
Section 1. Introduction
S-paths problem (µ-CEDP):
µ-CEDP:
Maximize
X
P ∈P
µ(sP , tP ) − a(P )
over all edge-disjoint sets P of S-paths in G.
Here sP , tP are the terminals which P connects, and a(P ) is the sum of edge-cost a(e)
over all edges e in P . The maximum value of µ-CEDP is denoted by val(µ; G, a).
The µ-CEDP is NP-hard since it includes the integer 2-commodity problem, which
is known to be NP-hard [10]. However, there is a few instance of terminal weights µ
for which µ-CEDP is tractable and admits a nice combinatorial min-max theorem.
Mader [31] established a combinatorial min-max relation for the case µ = 1 and
a = 0:
(
)
X
1
|δ G Xs | − κ ,
(1.1)
val(1; G, 0) = min
2
s∈S
where the minimum is taken over all disjoint node subsets Xs (s ∈ S) with s ∈ Xs .
Here δ G X denotes the set of edges joining
X and V \ X, and κ denotes the number
S
of connected components K in G − s∈S Xs such that |δ G V K| is odd. Mader [32]
extended (1.1) to openly node-disjoint S-paths packing; see [38] for a short proof.
Lovász [29] derived it from the matroid matching theory. Schrijver [39] formulated it
as a linear matroid matching problem. Thus maximum edge-disjoint S-paths can be
found in polynomial time by a linear matroid matching algorithm by Lovász.
Our main result is a complete characterization of terminal weights µ for which µCEDP is tractable and admits a combinatorial min-max theorem, extending Mader’s
edge-disjoint S-paths theorem. To state it, we need some notions. A tree metric is
a weight function that can be represented as distances among points in a weighted
tree. More precisely, a tree realization of a terminal weight µ is a triple (Γ, {ps }s∈S ; γ)
consisting of a tree Γ = (V Γ, EΓ ), a family {ps }s∈S ⊆ V Γ of nodes indexed by S,
and a positive γ > 0 such that
µ(s, t) = γdΓ (ps , pt ) (s, t ∈ S),
where dΓ denotes the shortest path metric of Γ (with respect to unit edge-length). A
terminal weight µ is called a tree metric if it has a tree realization. Here we can take
γ as 1/2 since µ is integer-valued.
A graph G = (V, E) with terminal set S is said to be inner Eulerian if every node
except terminals has even degree. An inner-odd-join is an edge subset F ⊆ E for
which G − F = (V, E \ F ) is inner Eulerian. For a tree Γ and a map ρ : V → V Γ , let
dΓ ρ denote a function on E defined by dΓ ρ (e) = dΓ (ρ(x), ρ(y)) for e = xy. Now we
are ready to state our main theorem:
Theorem 1.1. Suppose that µ is an integer-valued tree metric realized by (Γ, {ps }s∈S ;
1/2). Then we have
1 ρ
dΓ − a (E \ F ),
(1.2)
val(µ; G, a) = min max
2
EGRES Technical Report No. 2010-13
3
Section 1. Introduction
where the maximum is taken over inner-odd-joins F and the minimum is taken over
maps ρ : V → V Γ satisfying ρ(s) = ps for s ∈ S. Moreover there exists a polynomial
time algorithm to find edge-disjoint S-paths attaining the maximum.
Mader’s edge-disjoint S-path theorem (1.1) can be deduced from our theorem as
follows. Let µ = 1, and let Γ be the star with |S| leaves {ps }s∈S and center p0 . Then
(Γ, {ps }s∈S ; 1/2) realizes µ. Take a map ρ : V → V Γ with ρ(s) = psP
(s ∈ S) attaining
ρ
−1
the minimum. Let Xs := ρ (ps ) for s ∈ S. Then dΓ (E)/2 = s∈S |δ G Xs |. We
may assume that each node x ∈ Xs is reachable to s in the subgraph induced by Xs .
Otherwise replace ρ(x) by p0 for all nonreachable nodes x. Then dSΓ ρ is nonincreasing.
Let Y1 , Y2 , . . . , Yk be the node sets of connected components of G− s∈S Xs . Any innerodd-join F meets at least one edge in δ G Yi if |δ G Yi | is odd. For each i with |δ G Yi | odd,
take one edge from δ G Yi . By adding edges e = xy with ρ(x) = ρ(y),Pwe can extend it to
an inner-odd-join F ∗ of G. Then val(1; G, 0) = dΓ ρ (E \ F ∗ )/2 = ( s∈S |δ G Xs | − κ)/2.
Let us mention further results, relevances and applications.
Maximum edge-disjoint S-paths of minimum total cost. In 90’s, Karzanov
[23, 25] studied µ-CEDP for µ = p1 (p > 0), and gave a min-max theorem and
a combinatorial polynomial time algorithm for it. The proof sketch was given in
[25]. However the full proof [23] takes over 60 pages, is much complicated, and is
not published. So this paper includes a shorter and accessible proof of his result.
Moreover our min-max formula includes the dual half-integrality. Indeed, the inner
maximization is essentially a minimum-cost T -join problem. Then it can be dualized
by Seymour’s odd-cut packing theorem [41]. This dual half-integrality affirmatively
resolves an open question raised by Karzanov [25, Section 6, (2)] as the special case
of µ = p1; see Section 5
Tractability classification of terminals weights. Let us consider the cost-less
version (µ-EDP) of µ-CEDP:
X
µ-EDP:
Maximize
µ(sP , tP ) over all edge-disjoint sets P of S-paths in G.
P ∈P
A terminal weight µ is said to be tractable if µ-EDP is solved in polynomial time for
every graph containing terminal set S. The all-one weight 1 is tractable. By our
result, any tree metric is tractable. However, a slightly more general class of weights
is tractable. A weight µ is said to be a truncated tree metric if it can be represented
as distances among balls in a weighted tree, equivalently, there are a tree metric µ̄
and a nonnegative function (radius) r : S → R+ such that
(1.3)
µ(s, t) = max(µ̄(s, t) − r(s) − r(t), 0) (s, t ∈ S).
Corollary 1.2. Any truncated tree metric is tractable.
Indeed, suppose that µ is a truncated tree metric represented as (1.3) for a tree
metric µ̄ and r : S → R+ . Then µ-EDP reduces to µ̄-CEDP as follows. We may
EGRES Technical Report No. 2010-13
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Section 1. Introduction
assume that each terminal is incident to exactly one inner node (by super-terminal
argument). Define cost a by a(e) := r(s) if e joins a terminal s and a(e) := 0
otherwise. Then the value of an (s, t)-path P is µ̄(s, t) − a(s) − a(t), and hence any
optimal solution for µ̄-CEDP with this cost a is optimal to µ-EDP.
This result is best possible. Conversely we prove:
Theorem 1.3. If µ is not a truncated tree metric, then µ-EDP is NP-hard.
This hardness implies that tractable weights are exactly truncated tree metrics
unless P = N P .
Application: Approximating edge-disjoint S-paths for “near” tree metric.
Finding a good approximation for µ-EDP (with general µ) is a p
great challenge in the
area of approximation algorithms [42]; the current best is an O( |V |)-approximation
with due to Chekuri, Khanna and Shepherd [4]. The following observation suggests
some use of our result to the design of approximation algorithms for µ-EDP:
For a weight µ, if we can choose some tractable weight µ∗ such that
µ∗ ≤ µ ≤ αµ∗
for some α ≥ 1, then any optimal solution for µ∗ -EDP is an α-approximate
solution for µ-EDP.
In the case of metric weights, this problem—finding a good embedding into a tree
metric— falls into the area of the low-distortion embedding; see [30]. For two metric
spaces (X, d) and (Y, d0 ), a map φ : X → Y is an embedding with distortion α if
d0 (φ(x), φ(y)) ≤ d(x, y) ≤ αd0 (φ(x), φ(y)) for x, y ∈ X. Given a metric µ (metric
space (S, µ)), let α∗ denote the minimum distortion of an embedding into a tree
metric (the metric space on a tree). Finding a tree metric with minimum distortion
α∗ is known to be NP-hard. In the case where input metric µ is a graph metric,
Bădoiu, Indyk, and Sidiropoulos [2] gave a constant factor approximation algorithm
for it. Consequently we get a constant factor approximation algorithm for µ-EDP
when µ is a graph metric, and its distortion into tree metrics is fixed:
Corollary 1.4. Let µ be a graph metric with the minimum distortion α∗ into tree
metrics. Then there exists an O(α∗ )-approximation algorithm for µ-EDP.
The current best approximation factor is 6 due to [6]. So there is a 6α∗ -approximation algorithm for µ-EDP when µ is a graph metric. Unfortunately, the minimum
distortion of general metrics into tree metrics is Ω(|S|) [36], and Bartal’s probabilistic
embedding [1] does not work (since our problem is a “maximization” problem). We
hope that our results and techniques would provide tools to approximation algorithms
for µ-EDP/CEDP.
EGRES Technical Report No. 2010-13
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Section 1. Introduction
Multicommodity flows. A natural fractional relaxation of µ-EDP—µ-weighted
multiflow problem— has been well-studied in the literature. Hu [19] proved the existence of a half-integral optimal solution for the 2-commodity flow. Lovász [28] and
Cherkassky [5] independently proved the existence of a half-integral optimal solution
for all-one weight µ = 1. However the 3-commodity flow does not have such a property. Such phenomena of the (non-)existence of an optimal solution with bounded
denominator lead another kind of a weight classification problem, called the fractionality problem. The fractionality problem [21, 22] asks the classification of weights µ
for which every µ-weighted multiflow problem has a 1/k-integral optimal multiflow for
some integer k. Recent works [15, 16, 17] by Hirai gave a solution to the fractionality
problem. Also he gave a solution to the node-capacitated variation as: µ has bounded
fractionality for node-capacitated µ-weighted multiflow problems if and only if µ can
be represented as the distance between subtrees (not necessarily balls!) in a tree [18].
This work is much influenced by these developments.
Discrete minimax relation. A further fractional look is insightful. Via the TDI
argument, the existence of an optimal multiflow with bounded denominator implies
the existence of a dual optimal solution with bounded denominator, which represents a
combinatorial min-max relation. Moreover the half-integrality is sometimes improved
to be the integrality if the input graph is inner Eulerian. Indeed, it is known [26] that
if G is inner Eulerian and µ is an integer tree metric realized by (Γ, {ps }s∈S ; 1/2), then
the µ-weight multiflow problem has an integral optimal multiflow with value
1
min dρ (E),
ρ 2
where the minimum is taken over all maps ρ : V → V Γ with ρ(s) = ps for s ∈ S;
see Section 2.1. Since the complement of the union of edge-disjoint S-paths is an
inner-odd-join, for a general graph G (not necessarily inner Eulerian) we have
1
val(µ; G, 0) = max val(µ; G − F, 0) = max min dρ (E \ F ),
ρ 2
F
F
where the maximum is taken over all inner-odd-joins F . Therefore our result implies
a discrete minimax theorem; “max” and “min” are interchangeable. Note that the
weak duality of (1.2) can be seen by the general relation maxx∈X miny∈Y f (x, y) ≤
miny∈Y maxx∈X f (x, y).
Organization of the paper. The first goal is the proof of Theorem 1.1, and the
second goal is the proof of the hardness result (Theorem 1.3). We construct an
LP-relaxation of µ-CEDP/EDP by the following observation: the union of S-paths
is an inner Eulerian graph, and hence the complement is an inner-odd-join. Thus
val(µ; G, a) = maxF val(µ; G − F, 0) + a(E \ F ) holds. Next relax inner-odd-joins
into points in the convex hull of the incidence vector of inner-odd-joins (inner-oddjoin polytope), and relax edge-disjoint paths into multiflows in a natural way. Note
that the inner-odd-join polytope is a T -join polytope. So this LP-relaxation can be
EGRES Technical Report No. 2010-13
6
Section 1. Introduction
solved in polynomial time by the ellipsoid method [12]. This LP-relaxation was used
in [3, 25, 27] (implicitly or explicitly). We prove that if µ is a tree metric, then this
LP-relaxation has an integral optimal solution. This extends a result of Keijsper,
Pendavingh, and Stougie [27] for µ = 1 and a = 0. Although their proof assumes
Mader’s theorem (1.1), our proof does not, and is purely polyhedral, i.e., it consists only
of LP-based techniques. So our paper includes a new proof of Mader’s edge-disjoint
path theorem; it seems the first polyhedral proof.
In Section 2, we give preliminary arguments including the description of inner-oddjoin polytopes and tree-representation for a laminar family. In Section 3, we prove
the former part of Theorem 1.1, which consists of three steps. The first step is to
represent the dual of the LP-relaxation as a facility location problem on a tree. The
second step is to show that a dual optimum is always attained by a nice structured
combinatorial solution, and the optimal value coincides with RHS in the min-max
formula (1.2). In the third step, we analyze the stability of dual combinatorial solution
under edge deletion; we see an unexpected application of the polynomial uncrossing
process [25]. Then we prove by induction the existence of an integral optimal solution.
In Section 4, we prove polynomial solvability and NP-hardness. The polynomial
solvability is almost proved by the argument in the previous section. The proof of
the NP-hardness result is based on an interesting connection between a reduction to
integer 2-commodity flow and 4-point characterization of truncated tree metrics. As is
well known, a metric is a tree metric if and only if there is no 4-point subset {s, t, s0 , t0 }
(violating quadruple) satisfying µ(s, t) + µ(s0 , t0 ) > max{µ(s, t0 ) + µ(s0 , t), µ(s0 , t) +
µ(s, t0 )}. We prove a similar 4-point characterization for truncated tree metrics. From
violating quadruple s, t, s0 , t0 , we can reduce a version of integer 2-commodity flow
problem to µ|S 0 -EDP on four terminal S 0 = {s, t, s0 , t0 }. In Section 5, we discuss
related issues and raise future research directions.
Notation. Let R and R+ denote the sets of reals and nonnegative reals, respectively,
and let Z and Z+ denote the sets of integers and nonnegative integers, respectively.
For a graph Γ , let V Γ and EΓ denote the sets of nodes and edges, respectively. For
a set X, let RX and RX
+ denote the sets of functions from X and R and from X and
R+ , respectively. For a function f : X → R, let supp+ f and supp− f denote the sets
of elements x ∈ X with f (x) > 0 and f (x) < 0, respectively. Let f + , f − , and |f |
denote the functions defined by f + (x) = max{0, f (x)}, f − (x) = min{0, f (x)}, and
|f |(x) = max{−f (x), f (x)}, respectively. For a subset Y ⊆ X, let f (Y ) denote the
sum of f (y) over all y ∈ Y .
By a metric on a set V , we means a function d : V × V → R+ such that d(x, x) = 0,
d(x, y) = d(y, x), and triangle inequalities d(x, y) + d(y, z) ≥ d(x, z). For a subset
X ⊆ V , the cut metric δX defined by δX (x, y) = 1 if |X ∩ {x, y}| = 1 and δX (x, y) = 0
otherwise.
Let G = (V, E) be a graph with possible parallel edges and loops. For an edge e,
we write e = xy if e joins nodes x, y. For a node subset X ⊆ V , let δ G X denote
the set of edges joining
P X and V \ X. For two functions f, g on E, let f · g denote
the inner product e∈E f (e)g(e). We regard a metric d on V as a function on E by
d(e) := d(x, y) for e = xy ∈ E. In particular, δX · ξ = ξ(δ G X) for ξ ∈ RE .
EGRES Technical Report No. 2010-13
7
Section 2. Preliminaries
2
Preliminaries
Let G = (V, E) be an undirected graph with terminal set S ⊆ V , and let µ be a
terminal weight. Recall that an inner-odd-join is an edge subset F for which G − F
is inner Eulerian. Our starting point is the following relation:
(2.1)
val(µ; G, a) = max val(µ; G − F, 0) − a(E \ F ),
F
where the maximum is taken over all inner-odd-joins F in G. To see this, let P ∗ be
an edge-disjoint set of S-paths attaining val(µ; G, a), and let F ∗ be P
the complement of
∗
∗
the union of P . Then F is an inner-odd-join, and val(µ; G, a) = P ∈P ∗ µ(sP , tP ) −
a(E \ F ∗ ) ≤ val(µ; G − F ∗ , 0) − a(E \ F ∗ ). Conversely,P
for any inner-odd-join F and
any
P edge-disjoint set P of S-paths in G − F , we have P ∈P µ(sP , tP ) − a(E \ F ) ≤
P ∈P µ(sP , tP ) − a(P ) ≤ val(µ; G, a), where we use the nonnegativity of a for the
first inequality.
Motivated by this fact, we consider the following linear programming relaxation.
Let QG,S ⊆ RE denote the convex hull of the incidence vectors of all inner-odd-joins
in G. Let ΠG,S denote the set of all S-paths in G. Consider LP:
X
X
(2.2)
max
µ(sP , tP )f (P ) − a(E) +
a(e)ξ(e)
P ∈ΠG,S
subject to
ξ(e) +
e∈E
X
P ∈ΠG,S :e∈P
f (P ) ≤ 1 (e ∈ E),
ξ ∈ QG,S , f : ΠG,S → R+ .
The maximum value of this LP is denoted by val(µ; G, a). Obviously val(µ; G, a) ≤
val(µ; G, a). We are going to show that if µ is a tree metric, then this LP has an
integral optimal solution, i.e., val(µ; G, a) = val(µ; G, a).
2.1
Laminar locking theorem
In the case where G is inner Eulerian and a = 0, our problem becomes much easier; we
may put ξ = 0 in (2.2). So (2.2) coincides with the natural LP-relaxation of µ-EDP.
We will use the following fact as a base case of our inductive argument:
Lemma 2.1. If µ is a tree metric, G is inner Eulerian, and a = 0, then there exists
an integral optimal solution (ξ ∗ , f ∗ ) in (2.2) with ξ ∗ = 0.
This is a special case of the multiflow locking theorem due to Karzanov-Lomonosov
[26]. The multiflow locking theorem says that if a family F ⊆ 2S has no pairwise
crossing triple and G is inner Eulerian, then there exists an integral multiflow f being
simultaneously maximum (single-commodity) (A, S \ A)-flow for all X ∈ F. One can
easily see that
Psuch a multiflow f is a maximum multiflow with respect to a terminal
weight µ =
X∈F β(X)δX for any nonnegative weight β : F → R+ . As is wellknown, tree metric µ is the nonnegative sum of cut metrics for a laminar family; see
Section 2.2 below.
A simpler and shorter proof of the multiflow locking theorem is available at [11],
and a faster algorithm finding an integer optimal solution is given by [20].
EGRES Technical Report No. 2010-13
2.2
2.2
8
Trees and laminar family
Trees and laminar family
A metric-tree T is a metric space isometric to 1-dimensional contractible complex
endowed with the length metric. The metric function is denoted by dT . For a point
p ∈ T and nonnegative ≥ 0, let B(p, ) and B ◦ (p, ) denote the subsets of points
q ∈ T with dT (p, q) ≤ and dT (p, q) < , respectively. For two points p, q, let [p, q]
denote the set of points r with d(p, q) = d(p, r) + d(r, q), and let (p, q) := [p, q] \ {p, q}.
A vertex of T is a point q such that B ◦ (p, ) is not homeomorphic to an open segment
for every > 0. A metric-tree of two vertices is particularly called a segment.
Let Γ be a tree (in the graph-theoretical sense). From Γ we can construct a metrictree as follows. For each edge e = uv ∈ EΓ , consider segment Fe with twoSvertices
pe,u , pe,v and unit length dFe (pe,u , pe,v ) = 1. Then consider disjoint union e∈E Fe ,
and for each node u ∈ V Γ , identify points pe,u for all edge e incident to u; the image
of pe,u is denoted by pu . The resulting metric-tree is denoted by Γ̄ . Then (V Γ, dΓ )
isometrically embeds into (Γ̄ , dΓ̄ ) by v 7→ pv . So we can identify v and pv , and regard
as V Γ ⊆ Γ̄ .
Recall the tree representation of a laminar family; see [39, Section 13.4]. Two sets
X, Y ⊆ V are said be intersecting if X ∩ Y , X \ Y , and Y \ X are all nonempty.
A family of sets without intersecting pairs is called laminar. Let T be a metric-tree
with a specified point r ∈ T , called a root. Let ϕ : V → T be a map from a set V .
Let ET ,ϕ be the set of inclusion-maximal connected subsets ω ⊆ T not meeting the
image of ϕ, vertices, and root r. Each ω ∈ ET ,ϕ is an open segment. Let πω denote
its length, and let Xω denote the set of nodes x ∈ V such that ϕ(x) belongs to the
connected component of T \ ω not containing root r.
(2.3)
(1) {Xω | ω ∈ ET ,ϕ } is laminar.
P
(2) dT ϕ = ω∈ET ,ϕ πω δXω .
In particular, a tree metric is a nonnegative sum of cut metrics for a laminar family.
It is well-known that for any set function π : 2V → R+ having laminar support, there
exist a metric-tree T with root r and a map ϕ : V → T such that ET ,ϕ = supp+ π
and πω = π(Xω ) for ω ∈ ET ,ϕ ; see e.g.,[40]. In this case, we say “(T , ϕ) realizes π.”
2.3
T -joins and inner-odd-joins
In a graph G = (V, E) with even cardinality node subset T , a T -join is an edge subset
F ⊆ E so that odd(F ) = T , where odd(F ) is the set of nodes incident to odd number
of edges in F . See [39, Chapter 29] for T -joins. So an inner odd-join is exactly a T -join
when taking T as the set of nodes having odd degree in the graph G/S obtained by
identifying S into one node. Applying Edmonds-Johnson theorem [9] to our setting,
we see that QG,S coincides with the set of points ξ satisfying
(2.4)
0 ≤ ξ(e) ≤ 1
δX · ξ − 2ξ(F ) ≥ 1 − |F |
(e ∈ E),
(X ⊆ V \ S, F ⊆ δ G X, |δ G X \ F | : odd).
EGRES Technical Report No. 2010-13
2.4
9
Polynomial uncrossing process
The description of the up-hull QG,S + RE
+ is more simple and easier to handle, which
is given by the set of points ξ satisfying
(2.5)
ξ(e) ≥ 0
δX · ξ ≥ 1
(e ∈ E),
(X ⊆ V \ S, |δ G X| : odd).
A linear optimization over the T -join polytope with respect to a nonnegative objective
vector w is equivalent to that over its up-hull. In the case of general vector w, by
changing coordinate ξ(e) 7→ 1 − ξ(e) for e ∈ supp− w, a linear optimization over
the T -join polytope reduces to that over the up-hull of the T 4(odd(supp− (w)))-join
polytope; see [39, Section 29.1].
We apply it to our setting. For a vector w ∈ RE , define set function h[w] = hG [w]
on V \ S by
1 if |δ G X \ supp− w| is odd,
h[w](X) :=
(X ⊆ V \ S).
0 if |δ G X \ supp− w| is even,
Then we have the following:
(2.6)min{w · ξ | ξ ∈ QG,S }
= w− (E) + min {|w| · ξ | δX · ξ ≥ h[w](X) (X ⊆ V \ S)}


 X

X
= w− (E) + max
π(X)h[w](X) π(X)δX (e) ≤ |w|(e) (e ∈ E) .


X⊆V \S
X⊆V \S
We end this subsection with noting the following property of h[w]: For disjoint subsets
X, Y ⊆ V \ S we have
(2.7)
2.4
h[w](X ∪ Y ) ≡ h[w](X) + h[w](Y )
≡ |δ G X \ supp− w| + |δ G Y \ supp− w| mod 2.
Polynomial uncrossing process
In this subsetion, let h = hG [w] for simplicity. Motivated by (2.6), we consider the
following cut-packing program for v : E → R+ :
X
Min.
π(X)
X
s.t.
X
X
π(X)δX (e) ≤ v(e) (e ∈ E),
π : supp+ h → R+ .
Here the 0-1 set function h has the following property (the skew-supermodularity):
X, Y ∈ supp+ h
=⇒ X ∩ Y, X ∪ Y ∈ supp+ h or X \ Y, Y \ X ∈ supp+ h.
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Polynomial uncrossing process
We are particularily interested with a feasible soltion π whose support supp+ π is
laminar. Such a solution is called laminar.
Now we are given a arbitrary feasible function π. Consider the following process,
called an uncrossing process.
(0) If supp+ π is laminar, then stop.
(1) Choose an intersecting pair X, Y ∈ supp+ π and let α = min(π(X), π(Y )).
(2) Choose (X 0 , Y 0 ) ∈ {(X ∪ Y, X ∩ Y ), (X \ Y, Y \ X)} with X 0 , Y 0 ∈ supp+ h,
decrease π by α on X, Y and increase π by α on X 0 , Y 0 . Go to (0).
The step (2) is particularly called an uncrossing step. By an arbitrary choice of
X, Y, X 0 , Y 0 in each iteration, this process terminates after finitely many uncrossing
steps. So we can make any feasible solution laminar keeping feasility and without
decreasing the objective value. We say “uncross π” when we apply an uncrossing
process to π.
Karzanov [24] proved (for more general set systems) that by appropriate choice of
X, Y, X 0 , Y 0 we can uncross π so that the number of the uncrossing steps is polynomially bounded by for |V \ S| and | supp+ π|(≤ |2V \S |) for the initial π. We state it
as a suitable form for our purpose.
Theorem 2.2 ([24]). There exists a function p : Z+ → Z+ such that for every feasible
solution π we can uncross π in at most p(|V \ S|) uncrossing steps.
This result has an unexpected application about the stability of feasible solutions,
which we use at a key point (Lemma 3.13) of the whole proof.
Lemma 2.3. Let π be a laminar feasible solution. If π is not optimal, then for
arbitrary > 0 there exists a laminar feasible solution π ∗ having a smaller objective
value and satisfying
|π ∗ (X) − π(X)| < (X ∈ supp+ h).
The proof technique is also important for us. By convexity, for sufficiently small
> 0 there exists a feasible solution π 0 having a smaller objective value and satisfying
0
|π 0 (X) − π(X)| < 0
(X ∈ supp+ h).
Here π 0 is not necessarily laminar. Next apply Karzanov’s polynomial uncrossing
process to π 0 . Then we get a laminar feasible solution π ∗ having a smaller objective
value than that of π.
We show that π ∗ is also sufficiently close to π. Since 0 is sufficiently small, we have
supp+ π ⊆ supp+ π 0 . Therefore each uncrossing step chooses X, Y so that at least one
of X, Y does not belong to supp+ π. So the change in one uncrossing step is bounded
by the maximum value on supp+ h \ supp+ π. Moreover the number of uncrossing
steps is bounded by a function p(n) on n = |V \ S|. From this we have
∗
0
0
∗
0
|π(X) − π (X)| ≤ |π(X) − π (X)| + |π (X) − π (X)| ≤ +
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p(n)
X
k=1
2k 0 .
11
Section 3. Combinatorial min-max theorem
Since 0 > 0 was arbitrary, we may assume that |π(X) − π ∗ (X)| is sufficiently small
≤ .
3
Combinatorial min-max theorem
The goal of this section is to prove the min-max relation (1.2) in Theorem 1.1.
Throughout this section, G = (V, E) is an undirected graph with terminal set S ⊆ V ,
µ is a terminal weight on S, and a is an edge-cost.
For a set function h on V \ S, a metric-tree T with root r, a map ϕ : V → T with
ϕ(S) = {r}, let h ◦ (ϕ, T ) be defined by
X
h ◦ (ϕ, T ) :=
πω h(Xω ).
ω∈ET ,ϕ
3.1
Duality by tree location model
Suppose that µ is a tree metric realized by (Γ, {ps }s∈S ; 1). Our first task is to establish
the duality relation for (2.2) by a tree location model. We consider a triple (ρ, ϕ, T )
of a metric-tree T with root r, maps ρ : V → Γ̄ and ϕ : V → T . Regard (ρ, ϕ) as a
map from V to the product Γ̄ × T of two metric-trees. A triple (ρ, ϕ, T ) is said to be
feasible if it satisfies
(1) (ρ, ϕ)(s) = (ps , r) for s ∈ S,
(2) dT ϕ (e) ≤ |dΓ̄ ρ − a|(e) for e ∈ E.
For a feasible triple (ρ, ϕ, T ), let
τ G (ρ, ϕ, T ) := (dΓ̄ ρ − a)+ (E) − hG [dΓ̄ ρ − a] ◦ (ϕ, T ).
Theorem 3.1. val(µ; G, a) = min τ G (ρ, ϕ, T ), where the minimum is taken over all
feasible triples (ρ, ϕ, T ).
Corollary 3.2. val(µ; G, a) = min max(dΓ̄ ρ − a)(E \ F ), where the minimum is taken
over maps ρ : V → Γ̄ satisfying ρ(s) = ps for s ∈ S, and the maximum is taken over
all inner-odd-joins F .
The rest of this subsection is devoted to proving Theorem 3.1 and Corollary 3.2.
By substituting (2.4) to our LP (2.2), the dual LP is given by
X
Min.
l(E) + ˜l(E) −
πX,F (1 − |F |)
X,F
s.t.
l(P ) ≥ µ(sP , tP ) (P ∈ ΠG,S ),
X
l + ˜l −
πX,F (δX − 2χF ) ≥ a,
X,F
l, ˜l : E → R+ , πX,F ≥ 0,
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Duality by tree location model
where (X, F ) is taken over all pairs of node X ⊆ V \ S and edge set F ⊆ δ G X with
|δ G X \ F | odd, and χF is the incidence vector of F ⊆ E. Next fix l, and dualize
˜l, πX,F . Then we have
val(µ; G, a) = min max {(l − a)(E) − (l − a) · ξ}
l
ξ∈QG,S
= min
l
max
F :inner-odd-join
(l − a)(E \ F ),
where the minimum is taken over all l : E → R+ with l(P ) ≥ µ(sP , tP ) for P ∈ ΠG,S .
Let g G : RE → R be defined by
g G (l) :=
(3.1)
max
F :inner-odd-join
(l − a)(E \ F ) (l ∈ RE ).
Then g G is monotone nondecreasing. Indeed, for l ≤ l0 , take an inner-odd-join F ∗
with g G (l) = (l − a)(E \ F ∗ ). Then (l − a)(E \ F ∗ ) ≤ (l0 − a)(E \ F ∗ ) ≤ g G (l0 ).
Therefore we can replace l by the shortest path metric dG,l on G with respect to l.
Hence val(µ; G, a) is equal to the minimum value of the following convex optimization
over metrics:
(3.2)
Min.
s.t.
g G (d)
d: metric on V ,
d(s, t) ≥ µ(s, t) (s, t ∈ S).
Here we claim:
Claim 3.3. For any metric d on V feasible to (3.2), there exists a map ρ : V → Γ̄
such that ρ(s) = ps (s ∈ S) and dΓ̄ ρ (x, y) ≤ d(x, y) for x, y ∈ V .
This technique was used in [16, 18]; we give the proof for completeness.
Proof. Let V = {x1 , x2 , . . . , xn } and S = {x1 , x2 , . . . , xk }. Define ρ : V → Γ̄ recursively by
p xi
if i ≤ k,
T
ρ(xi ) :=
an arbitrary point in j<i B(ρ(xj ), d(xj , xi )) if i > k.
T
We show by induction that j<i B(ρ(xj ), d(xj , xi )) is nonempty, i.e., ρ is well-defined.
If true, then ρ(xi ) ∈ B(ρ(xj ), d(xj , xi )) implies dΓ̄ ρ (xi , xj ) ≤ d(xi , xj ). Recall that
any collection of subtrees has Helly property. So it suffices to show the pairwise
nonempty intersection. Here two balls B(p, r) and B(p0 , r0 ) intersects if and only if
dΓ̄ (p, p0 ) ≤ r + r0 . Therefore B(ρ(xj ), d(xi , xj )) ∩ B(ρ(xj 0 ), d(xi , xj 0 )) 6= ∅ follows from
d(xi , xj ) + d(xi , xj 0 ) ≥ d(xj , xj 0 ) ≥ dΓ̄ ρ (xj , xj 0 ), where the last inequality follows from
the induction.
Since g G is monotone nondecreasing, we get
(3.3)
val(µ; G, a) = min max {(dΓ̄ ρ − a)(E) − (dΓ̄ ρ − a) · ξ},
ξ∈QG,S
= min max(dΓ̄ ρ − a)(E \ F ),
F
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Dual integrality
where the minimum is taken over all maps ρ : V → Γ̄ satisfying ρ(s) = ps for s ∈ S.
Thus we have Corollary 3.2.
By (2.6), val(µ; G, a) is also equal to the minimum value of the following problem:
X
Min.
(d − a)+ (E) −
(3.4)
π(X)hG [d − a](X)
X⊆V \S
s.t.
X
X⊆V \S
π(X)δX (e) ≤ |d − a|(e) (e ∈ E)
d(s, t) ≥ µ(s, t) (s, t ∈ S),
d: metric on V , π : 2V \S → R+ .
For a feasible solution (d, π) in (3.4) let τ G (d, π) = τ (d, π) denote its objective value.
Let (ρ, ϕ, T ) be a feasible triple. Define π ϕ,T : 2V \S → R+ by
πω if X = Xω for ω ∈ ET ,ϕ ,
ϕ,T
(3.5)
π (X) =
(X ⊆ V \ S).
0 otherwise,
P
ρ
ϕ,T
By dT ϕ =
) is feasible to (3.4) and
ω∈Eϕ,T πω δXω and condition (2), (dΓ̄ , π
τ (d, π) = τ (ρ, ϕ, T ). Conversely, by (3.3) we can take an optimal solution (d, π)
in (2.6) so that d = dΓ̄ ρ for a map ρ : V → Γ̄ satisfying ρ(s) = ps for s ∈ S.
By the uncrossing process, we can make π so that supp+ π is laminar; the uncrossing operation keeps the feasibility and the objective value of (d, π), So there are
a metric-tree
P T with root r and aρ map ϕ : V → T realizing π. Then we have
ϕ
dT (e) = X⊆V \S π(X)δX (e) ≤ |dΓ̄ − a|(e) for e ∈ E. This means that (ρ, ϕ, T ) is
feasible triple with τ (ρ, ϕ, T ) = τ (d, π).
3.2
Dual integrality
The next task is to prove that the minimum of τ G (·) is attained by some structured
combinatorial solution. A feasible triple (ρ, ϕ, T ) is said be bipartite if T = T̄ for
some graph-theoretical tree T , ϕ(V ) ⊆ V T , ρ(V ) ⊆ V Γ , and dΓ̄ ρ + dT ϕ is evenvalued. Consider the product graph Γ × T of two trees Γ and T , which is also a
bipartite graph. From the last condition, the image of (ρ, ϕ) belongs to one of the
color classes (containing (ps , r)); see Figure 1.
Theorem 3.4. Suppose that µ is an even-valued tree metric realized by (Γ, {ps }s∈S ; 1),
and edge-cost a is even-valued. Then there exists a bipartite feasible triple (ρ, ϕ, T )
with val(µ; G, a) = τ G (ρ, ϕ, T ).
In particular, the image of ρ belongs to V Γ . This implies:
Corollary 3.5. Under the same assumption, we have
val(µ; G, a) = min max(dΓ ρ − a)(E \ F ),
where the maximum is taken over all inner-odd-join F and the minimum is taken over
all maps ρ : V → V Γ with ρ(s) = ps for s ∈ S.
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Dual integrality
T
T
r
ps
pu
pt
Figure 1: Product of two trees
The remainder of this subsection is devoted to the proof of Theorem 3.4. Let
(ρ, ϕ, T ) be a feasible triple. For points p, q ∈ T , we write p q if [r, p] ⊆ [r, q], and
write p ≺ q if [r, p] ⊂ [r, q] (proper inclusion). We endow T with the partial order
by this relation. We may assume that (ρ, ϕ, T ) is obtained from a rational-valued
solution (d, π) in (3.4). Then dΓ̄ ρ is rational-valued, and πω is rational for ω ∈ ET ,ϕ ,
and thus dT ϕ is also rational-valued. We may assume that they are multiples of some
positive rational ≤ 1/2. Let {R, B} be the color classes of (bipartite graph) Γ .
Since µ is even-valued, we may assume {ps }s∈S ⊆ B. For a point p in Γ̄ \ V Γ , there
is a unique edge uv in Γ with p ∈ [u, v]. Suppose (u, v) ∈ R × B (w.l.o.g.). Let p→B,
denote the unique point p0 with d(p, v) = d(p, p0 ) + , and let p→R, denote the unique
point p0 with d(p, u) = d(p, p0 ) + . Let Xq , Xq, , and Xq,≺ denote the sets of nodes x
with q = ϕ(x), q ϕ(x), and q ≺ ϕ(x), respectively.
Let us start the proof of Theorem 3.4. We may assume that
(3.6)
h[dΓ̄ ρ − a](Xω ) = 1 (ω ∈ ET ,ϕ ).
We are going to construct a bipartite feasible triple (ρ∗ , ϕ∗ , T ∗ ) with τ (ρ∗ , ϕ∗ , T ∗ ) ≤
τ (ρ, ϕ, T ). For notational simplicity, we often denote dT , dΓ̄ simply by d (if no confusion occurs), such as dϕ , dρ , d(ρ(x), b). An -point is a point q in T so that dT (q, p)
is a multiple of for some (every) vertex p in T . For an -point q ∈ T , let C = Cq
be the set of connected components of T \ B ◦ (q, ). Let C0 denote the (unique) connected component in C containing r when q 6= r. Consider an -point q ∈ T with the
following properties:
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Dual integrality
C ∈ C B , I B (∆C = 0)
C ∈ C R , I R (∆C = 0)
T
∆C
q
(b, q)
(b, q)
b
b
Γ
Γ
Figure 2: Drawing grids in C × Γ̄ for C ∈ C \ {C0 }
(3.7)
(i) ρ(Xq,≺ ) ⊆ V Γ .
(ii) For ω ∈ ET ,ϕ , πω is an integer if ω belongs to a connected component
in C \ {C0 }.
(iii) For x, y ∈ Xq,≺ , (dρ + dϕ )(x, y) is an even integer if ϕ(x) and ϕ(y)
belong to the same connected component in C \ {C0 }.
This means that the restriction of (ρ, ϕ, T ) to each component in C \ {C0 } is bipartite.
Such a point exists since we can take q as any maximal point. A minimal point with
this property is called a critical point. Take a critical point q ∈ T . We try to move
critical points toward r.
Fix b ∈ B. For a component C ∈ C \ {C0 }, take a node x with ϕ(x) ∈ C; it exists
by (3.6). By condition (3.7), we see that d(q, ϕ(x)) − bd(q, ϕ(x))c and the parity of
d(b, ρ(x)) + bd(q, ϕ(x))c are determined independently on the choice of x. We denote
them by ∆C and DC , respectively. Partition C \ {C0 } according to ∆C , DC as:
CB
CR
IB
IR
:=
:=
:=
:=
{C
{C
{C
{C
| ∆C
| ∆C
| ∆C
| ∆C
6= 0,
6= 0,
= 0,
= 0,
DC
DC
DC
DC
= even},
= odd},
= even},
= odd}.
Figure 2 illustrates a portion of C × Γ̄ for C ∈ C \ {C0 }. We can draw a grid on
C × Γ̄ so that its vertices are points having even distance-sum dΓ̄ + dT from some
(every) image of (ρ, ϕ) in C × Γ̄ . We try to extend these partial grids to the whole
grid; readers may keep in mind this intuition for the subsequent arguments.
Let X B be the set of nodes x ∈ Xq, such that d(b, ρ(x)) + d(q, ϕ(x)) is even.
Equivalently X B consists of nodes x ∈ Xq with ρ(x) ∈ B and nodes x ∈ Xq,≺
with ϕ(x) ∈ C ∈ I B . Similarly, let X R be the set of nodes x ∈ Xq, such that
d(b, ρ(x)) + d(q, ϕ(x)) is odd. For C ∈ C let qC denote the (unique) point in C
meeting B(q, ) (in T ).
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Dual integrality
IB
q
qC0
T
noncritical
q
X B = ∅ at q
q
ǫ
ǫ
qC0
qC0
T ′ : h(X B ) = 0
T ′ : h(X B ) = 1
Figure 3: Tree modification I
We first consider the case q 6= r. Suppose X B 6= ∅. We modify (ϕ, T ) into (ϕ0 , T 0 )
as follows. First let ϕ0 (x) := ϕ(x) for x ∈ V \ Xq .
Suppose h[dρ − a](X B ) = 0, implying h[dρ − a](Xq, \ X B ) = 1 by (2.7) and (3.6).
Next delete open ball B ◦ (q, ) from T . Let C be the set of the resulting connected
components. As above, partition C into {C0 } ∪ C B ∪ C R ∪ I B ∪ I R . For each C ∈ I B ,
join qC and qC0 by a segment with length . Add a new point q. For each C ∈ C \ I B ,
join q and qC by a segment of length . Now we obtain a new tree T 0 and obtain
a map ϕ0 : V \ Xq → T 0 . We extend ϕ0 to V → T 0 by defining ϕ0 (x) := qC0 for
x ∈ X B ∩ Xq and ϕ0 (x) := q for x ∈ Xq \ X B . See the middle of Figure 3. We claim
Claim 3.6. (ρ, ϕ0 , T 0 ) is feasible, and τ (ρ, ϕ0 , T 0 ) = τ (ρ, ϕ, T ).
Proof. Consider edge e = xy ∈ E. It suffices to consider the case where x ∈ X B and
0
y ∈ Xq, \ X B ; for other cases we have dϕ (e) ≤ dϕ (e), and thus the feasibility keeps.
0
Then dϕ (e) = dϕ (e) + . So it suffices to show dϕ (e) < |dρ (e) − a(e)|. This follows
from the fact that a(e) is even and dρ (e) + dϕ (e) is not even. The latter part follows
from: h ◦ (ϕ0 , T 0 ) − h ◦ (ϕ, T ) = {h(Xq, \ X B ) − h(Xq, )} = 0, where we denote
h[dρ − a] by h, and we use (2.7) with h(Xq, ) = 1 (3.6) and h(X B ) = 0.
Let (ρ, ϕ, T ) ← (ρ, ϕ0 , T 0 ). Then X B becomes empty at q. Suppose that h[dρ −
a](X B ) = 1, which implies h[dρ − a](Xp, \ X B ) = 0. Therefore we can apply the
above-described process with changing the roles of X B and Xp, \X B . See the right of
Figure 3. Then we get a feasible triple (ρ, ϕ0 , T 0 ). Let (ρ, ϕ, T ) ← (ρ, ϕ0 , T 0 ). Suppose
X R 6= ∅. Then apply the above-described procedure by changing roles of B and R.
So consider the case where X B ∪ X R = ∅, which implies I B ∪ I R = ∅. Here
we construct two modifications (ρ0 , ϕ0 , T 0 ) and (ρ00 , ϕ00 , T 00 ), as follows. First let
(ρ0 , ϕ0 )(x) := (ρ, ϕ)(x) for x ∈ V \ Xq . Next delete open ball B ◦ (q, ) from T . Let
C = {C0 } ∪ C B ∪ C R be the set of resulting connected components. For each C ∈ C R ,
join qC and qC0 by a segment with length 2. For each C ∈ C B , identify (glue) qC and
qC0 . For each x ∈ Xq , let ϕ0 (x) := qC0 and let ρ0 (x) := (ρ(x))→B, (well-defined by
X B ∪ X R = ∅). Let (ρ0 , ϕ0 , T 0 ) be the resulting triple. See Figure 4 for the construction of T 0 . The second triple (ρ00 , ϕ00 , T 00 ) is obtained by changing roles of B and R.
Now we show:
Claim 3.7.
(1) Both (ρ0 , ϕ0 , T 0 ) and (ρ00 , ϕ00 , T 00 ) are feasible.
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Dual integrality
CR
CB
q
ǫ q
C0
T
2ǫ
qC0
qC0
T \ B ◦(q, ǫ)
T′
Figure 4: Tree modification II
0
00
(2) supp± (dρ − a) ⊆ supp± (dρ − a) and supp± (dρ − a) ⊆ supp± (dρ − a).
(3) Either τ (ρ0 , ϕ0 , T 0 ) ≤ τ (ρ, ϕ, T ) or τ (ρ00 , ϕ00 , T 00 ) ≤ τ (ρ, ϕ, T ) holds.
Proof. (1). It suffices to show the feasibility of (ρ0 , ϕ0 , T 0 ). Take an edge e = xy ∈ E
0
0
with (ρ, ϕ)(x) 6= (ρ, ϕ)(y). We verify condition (∗) dϕ (e) ≤ |dρ (e) − a(e)| for the case
where (i) ϕ(x) = q ≺ ϕ(y) ∈ C ∈ C R , and (ii) ϕ(x) ∈ C ∈ C R , ϕ(y) ∈ C 0 ∈ C R , and
C 6= C 0 . For other cases, the condition (∗) is obviously fulfilled.
0
Case (i): ϕ(x) = q ≺ ϕ(y) ∈ C ∈ C R . In this case, dϕ (e) = dϕ (e) + . Suppose
ρ(x) ∈ [u, v] for u ∈ B and v ∈ R. For r, r0 ∈ [, 1 − ], dϕ (e) = bd(ϕ(y), q)c + r, and
either dρ (e) = d(ρ(y), v) + r0 or dρ (e) = d(ρ(y), u) + r0 .
0
Suppose dρ (e) − a(e) > 0. If dρ (e) = d(ρ(y), v) + r0 , then dρ (e) = dρ (e) + and
0
hence the feasibility keeps. So suppose dρ (e) = d(ρ(y), u) + r0 ; then dρ (e) = dρ (e) − .
It suffices to show
(3.8)
dϕ (e) − dρ (e) + 2 ≤ −a(e).
Let N := d(ρ(y), u) − bd(ϕ(y), q)c. Since u ∈ B and ϕ(y) ∈ C ∈ C R , N is an odd
integer. From dϕ (e) − dρ (e) = N + r − r0 and r, r0 ∈ [, 1 − ], we have
N − 1 + 2 ≤ dϕ (e) − dρ (e) ≤ N + 1 − 2.
Since a(e) is even, we have (3.8). The argument for the case dρ (e)−a(e) < 0 is similar.
0
Case (ii) ϕ(x) ∈ C ∈ C R , ϕ(y) ∈ C 0 ∈ C R , and C 6= C 0 . In this case, dρ (e) = dρ (e)
0
and dϕ (e) = dϕ (e) + 2. We show
(3.9)
dϕ (e) + 2 ≤ |dρ (e) − a(e)|.
For r, r0 ∈ [, 1 − ], we have dϕ (e) = bd(ϕ(x), q)c + bd(ϕ(y), q)c + r + r0 . Also for
an even integer N , we have dρ (e) = N + d(ρ(x), b) + d(ρ(y), b). By a similar parity
argument as above, we have (3.9).
0
(2). Take an edge e = xy ∈ E. We show that e ∈ supp± (dρ − a) implies e ∈
supp± (dρ − a). We may assume that ρ(x) = q. Suppose ρ(y) 6= q. Then |dρ − a|(e) ≥
0
dϕ (e) ≥ , and dρ (e) − dρ (e) ∈ {, −}. Then the claim follows.
0
0
So suppose ϕ(x) = ϕ(y) = q. Then dϕ (e) = dϕ (e) = 0, and dρ (e) − dρ (e) =
0
{2, 0, −2}. If dρ (e) = a(e) (even), then dρ (e) = dρ (e); ρ(x) and ρ(y) move toward
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Dual integrality
0
the same direction on a path of Γ̄ . So we may assume dρ (e) 6= a(e), and dρ (e)−dρ (e) =
{2, −2}. We can take p, p0 ∈ V Γ such that dρ (x, y) = d(ρ(x), p)+d(p, p0 )+d(p0 , ρ(y)),
and d(ρ(x), p), d(p0 , ρ(y)) ∈ [, 1 − ]. Then necessarily p and p0 belong to the same
color class, and hence N := d(p, p0 ) is even. Thus
N + 2 ≤ dρ (e) ≤ N + 2 − 2.
Since a(e) and N are even, dρ (e) > a(e) implies dρ (e) − 2 ≥ a(e), and dρ (e) < a(e)
0
implies dρ (e) + 2 ≤ a(e). Thus e ∈ supp± (dρ − a) implies e ∈ supp± (dρ − a).
0
(3). We first compare (dρ − a)+ (E) and (dρ − a)+ (E). For a node x ∈ V with
ρ(x) 6∈ V Γ , define disjoint subsets δ B x, δ R x of δ G x by the following way. We may
assume ρ(x) ∈ [u, v] for uv ∈ EΓ with u ∈ B, v ∈ R. Let KxB and KxR be the
connected components of Γ̄ \ {ρ(x)} containing u and v, respectively. Then let
δ B x := {e ∈ δ G x | e joins a node y with ρ(y) ∈ KxB },
δ R x := {e ∈ δ G x | e joins a node y with ρ(y) ∈ KxR }.
Then we have
(3.10)
0
((dρ − a)+ − (dρ − a)+ )(E)
0
= (dρ − dρ )(E \ supp− (dρ − a))
X
|δ R x \ supp− (dρ − a)| − |δ B x \ supp− (dρ − a)|
= x∈Xq
00
= ((dρ − a)+ − (dρ − a)+ )(E).
0
Here we use supp− (dρ − a) ⊆ supp− (dρ − a) for the first equality. To see the second
equality, consider an edge e ∈ E \ (supp− dρ − a) joining Xq ; other edges do not
contribute. Then either dρ (e) > 0 or dρ (e) = dϕ (e) = 0 by a(e) ≥ 0. We may assume
0
the former case; for the latter case obviously dρ (e) = dρ (e). Suppose e = xy and
x ∈ Xq . So e ∈ δ R x ∪ δ B x. By the case-by-case analysis, we see that dρ (e) increases
by if e ∈ δ R x and y 6∈ Xq , decreases by if e ∈ δ B x and y 6∈ Xq , increases by 2
if e ∈ δ R x ∩ δ R x and y ∈ Xq , decreases by 2 if e ∈ δ B x ∩ δ B x and y ∈ Xq , and
keeps otherwise. By a simple counting we see the second equality; the third equality
is obtained by changing the roles of R and B.
0
Next we compare h[dρ −a]◦(ϕ, T ) and h[dρ −a]◦(ϕ0 , T 0 ). For C ∈ C, let ω C denote
the (unique) open segment in ET ,ϕ containing (q, qC ). By construction, we have
(3.11)
 T ,ϕ
 π (X) − if X = XωC for C ∈ C B ∪ {C0 },
T 0 ,ϕ0
π T ,ϕ (X) + if X = XωC for C ∈ C R ,
(X ⊆ V \ S).
π
(X) =
 T ,ϕ
π (X)
otherwise,
Moreover it holds that
(3.12)
0
h[dρ − a](X) = h[dρ − a](X) (X ⊆ V \ S : π T
0
0
0 ,ϕ0
0
(X) > 0).
0
0
Indeed, take an edge e ∈ δ G X for π T ,ϕ (X) > 0. Then dϕ (e) ≥ π T ,ϕ (X) > 0, and
0
0
0 0
|dρ − a|(e) ≥ dϕ (e) > 0. By supp+ π T ,ϕ ⊆ supp+ π T ,ϕ , also |dρ − a|(e) ≥ dϕ (e) > 0
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0
0
holds. Since |dρ (e) − dρ (e)| ≤ by dϕ (e) > 0, the signs of (dρ − a)(e) and (dρ − a)(e)
must be same.
By (3.11) and (3.12) we have
0
h[dρ − a] ◦ (ϕ0 , T 0 ) − h[dρ − a] ◦ (ϕ, T ) = {|C R | − |C B | − 1}.
(3.13)
00
Similarly we have h[dρ − a] ◦ (ϕ00 , T 00 ) − h[dρ − a] ◦ (ϕ, T ) = {|C B | − |C R | − 1}. Let
∆0 := {τ (ρ0 , ϕ0 , T 0 ) − τ (ρ, ϕ, T )}/ and ∆00 := {τ (ρ00 , ϕ00 , T 00 ) − τ (ρ, ϕ, T )}/. Then we
have
∆00 = −∆0 + 2.
Therefore it suffices to show that ∆0 is an even integer. By (3.10) and (3.13), we have
X
∆0 ≡
(|δ R x \ supp− (dρ − a)| + |δ B x \ supp− (dρ − a)|) + (|C B | + |C R |) + 1
x∈Xq
≡
≡
X
x∈Xq
X
x∈Xq
ρ
|δ G x \ supp− (dρ − a)| +
ρ
h[d − a](x) +
X
x∈Xq,≺
≡ h[d − a](Xq, ) + 1 ≡ 0
X
C∈C\{C0 }
h[dρ − a](XωC ) + 1
ρ
h[d − a](x) + 1
mod 2.
In the second equality, we use the observation that for an edge e = xy ∈ E\supp− (dρ −
a) if e belongs to δ G x \ (δ B x ∪ δ R x) for x ∈ Xq , then y ∈ Xq and e belongs to
δ G y \ (δ B y ∪ δ R y) since dρ (e) = dϕ (e) = 0. Also we use (2.7) and h[dρ − a](XωC ) = 1
for C ∈ C by (3.6).
Let (ρ, ϕ, T ) ← (ρ0 , ϕ0 , T 0 ) if τ (ρ0 , ϕ0 , T 0 ) ≤ τ (ρ, ϕ, T ), and let (ρ, ϕ, T ) ← (ρ00 , ϕ00 ,
T 00 ) otherwise. We can repeat this process (in finitely many steps) until r becomes
critical.
So we consider the case where q = r. The argument is essentially same (or simpler).
Define ρ0 : V → Γ̄ by ρ0 (x) := (ρ(x))→B, for x ∈ Xq \ (X B ∪ X R ), and ρ0 (x) := ρ(x)
for x ∈ X B ∪ X R . Also define ϕ0 (x) := ϕ(x) for x ∈ V \ Xq . Next delete open ball
B ◦ (q, ) from T . Let C = C B ∪ C R ∪ I B ∪ I R be the set of the resulting connected
component in T . Add a new root r. For all C ∈ I B ∪I R , join qC and r by a segment of
length . For each C ∈ C B , join qC and r by a segment of length 2. For each C ∈ C R ,
identify (glue) qC and r. Define ϕ0 (x) = r for x ∈ Xq . Then we obtain a triple
(ρ0 , ϕ0 , T 0 ). The second triple (ρ00 , ϕ00 , T 00 ) is obtained by changing the roles of B and
R. By a similar analysis in Claim 3.7, both (ρ0 , ϕ0 , T 0 ) and (ρ00 , ϕ00 , T 00 ) are feasible,
and τ (ρ0 , ϕ0 , T 0 ) − τ (ρ, ϕ, T ) = −τ (ρ00 , ϕ00 , T 0 ) + τ (ρ, ϕ, T ). Let (ρ, ϕ, T ) ← (ρ0 , ϕ0 , T 0 )
if τ (ρ0 , ϕ0 , T 0 ) ≤ τ (ρ, ϕ, T ) and let (ρ, ϕ, T ) ← (ρ00 , ϕ00 , T 00 ) otherwise. Increase to
its multiple until X B ∪ X R increases. So we can repeat it until Xr, = X B ∪ X R .
If X R = ∅, then the current (ρ, ϕ, T ) is bipartite, as required. So suppose X R 6= ∅.
Delete open ball B ◦ (r, ) from T . Then C = I R ∪ I B . Consider a segment of length
1. Let r, q denote its vertices. For each C ∈ C B , join qC and r by a segment of length
. For each C ∈ C R , join qC and q by a segment of length . For x ∈ Xq , define
ϕ0 (x) := r if x ∈ X B , and ϕ0 (x) := q if q ∈ X B . The resulting (ρ, ϕ0 , T 0 ) is feasible.
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Indeed, if e join X B and X R , then dϕ (e) and |dρ (e) − a(e)| have different parity, and
this implies dϕ (e) + 1 ≤ |dρ (e) − a(e)|. From this we see the feasibility. Also (ρ, ϕ0 , T 0 )
is bipartite, and τ (ρ, ϕ0 , T 0 ) = τ (ρ, ϕ, T ) − h[dρ − a](X B ), as required.
Our construction includes the following important consequence.
Lemma 3.8. In the construction above, if (dΓ̄ ρ + dT ϕ )(x, y) ≤ L for an even integer
∗
∗
L, then (dΓ̄ ρ + dT ∗ ϕ )(x, y) ≤ L.
Proof. It suffices to verify that for each change (ρ, ϕ, T ) ← (ρ0 , ϕ0 , T 0 ) above, (dρ +
0
0
dϕ )(x, y) ≤ L implies (dρ + dϕ )(x, y) ≤ L. This has already done. Add an edge e
joining x and y with even cost a(e) = L. Then (ρ, ϕ, T ) is feasible to the new graph,
and then so does (ρ0 , ϕ0 , T 0 ) with property (2) in Claim 3.7. In any change, we have
0
0
(dρ + dϕ )(x, y) ≤ a(e) = L.
3.3
Primal integrality
The last step proves the existence of an integral optimal solution in (2.2).
Theorem 3.9. Suppose that µ is a tree metric. Then there exists an integral optimal
solution in (2.2). In particular, val(µ; G, a) = val(µ; G, a).
Together with Corollary 3.5, this establishes the min-max formula (1.2). The proof
of Theorem 3.9 is based on an inductive approach by edge-deletion. Take an edge
e ∈ E, consider G − e. As is expected, we have
(3.14)
val(µ; G − e, a) ≤ val(µ; G, a).
To see this, take any optimal solution (ξ, f ) in (2.2) for G−e. Extend ξ to ξ¯ : E → R+
¯ f ) is feasible to (2.2) for G. In particular,
by defining ξ(e) := 1. The resulting (ξ,
if (3.14) holds with equality and there exists an integral optimal solution in (2.2) for
G − e, then there also exists an integral optimal solution in (2.2) for G.
An edge e ∈ E is called deletable if val(µ; G − e, a) = val(µ; G, a). Our goal is to
prove a criterion of deletable edges.
Proposition 3.10. Let (ξ, f ) be an optimal solution in (2.2). For an edge e ∈ E, if
ξ(e) > 0, then e is deletable.
Assuming this lemma, we complete the proof of Theorem 3.9. We use the induction
on the number of edges. Take any optimal solution (ξ, f ) in (2.2). Suppose ξ = 0.
Then val(µ; G, a) = val(µ; G, 0) − a(E). Moreover G is necessarily inner Eulerian. By
the
P laminar locking theorem, we can take an edge-disjoint set P ofPS-paths so that
since P ∈P µ(sP , tP ) −
P ∈P µ(sP , tP ) = val(µ; G, 0). Then P attains the maximum
P
a(P ) ≤ val(µ; G, a) ≤ val(µ; G, a) = val(µ; G, 0) − a(E) ≤ P ∈P µ(sP , tP ) − a(P ). So
suppose that there is an edge e ∈ E with ξ(e) > 0. By the lemma, e is deletable. By
induction together with construction above, there exists an integral optimal solution.
The reminder of this subsection is devoted to proving Proposition 3.10. Without loss
of generality, we assume that µ is an even-valued tree metric realized by (Γ, {ps }s∈S ; 1),
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edge-cost a is even-valued, and
for each vertex p ∈ V Γ of degree one, there is s ∈ S with p = ps .
(3.15)
We need three lemmas. The first lemma is the parity of τ :
Lemma 3.11. If a feasible triple (ρ, ϕ, T ) is bipartite, then τ G (ρ, ϕ, T ) is an even
integer.
Proof. We use the induction on the number of edges; if E = ∅ then the statement
is obvious. Take an edge e = xy ∈ E. (ρ, ϕ, T ) is feasible to G − e. Then (∗)
hG−e [dρ − a](Xω ) 6= hG [dρ − a](Xω ) if and only if e ∈ δ G Xω \ supp− (dρ − a). Therefore,
if e ∈ supp− (dρ − a), then obviously τ G (ρ, ϕ, T ) = τ G−e (ρ, ϕ, T ). So suppose e 6∈
supp− (dρ − a). From (∗) we see dϕ (e) ≡ hG−e [dρ − a] ◦ (ϕ, T ) − hG [dρ − a] ◦ (ϕ, T )
mod 2. Also dρ (e) ≡ (dρ − a)+ (e) mod 2. Since dρ (e) ≡ dϕ (e) mod 2, we have
τ G−e (ρ, ϕ, T ) ≡ τ G (ρ, ϕ, T ) mod 2, as required.
The second lemma is a complementary slackness.
Lemma 3.12. Let (ξ, f ) be a feasible solution in (2.2) and let (ρ, ϕ, T ) be a feasible
triple. Then (ξ, f ) and (ρ, ϕ, T ) are both optimal if and only if the following four
conditions hold:
P
(a) For each e ∈ E with dρ (e) > 0 we have ξ(e) + P ∈ΠG,S ,e∈P f (P ) = 1.
(b) For each e ∈ E with dϕ (e) < |dρ − a|(e) we have
1 if e ∈ supp− (dρ − a),
ξ(e) =
0 otherwise.
(c) For each ω ∈ ET ,ϕ , we have
ξ(δXω \ supp− (dρ − a)) + (1 − ξ)(δXω ∩ supp− (dρ − a)) = h[dρ − a](Xω ).
(d) For each P ∈ ΠG,S with f (P ) > 0 we have dρ (P ) = µ(sP , tP ).
Proof. This follows from direct calculations:
(dρ − a)+ (E) − h[dρ − a] ◦ (ϕ, T ) −
=
X
e∈E
dρ (e)(1 − ξ(e) − f (e)) +
X
+
X
P ∈ΠG,S
P ∈ΠG,S
µ(sP , tP )f (P ) + (a − ξ)(E)
f (P ){dρ (P ) − µ(sP , tP )}
ρ
e∈E\supp− (dρ −a)
+
X
X
ξ(e)(|d − a| − dϕ )(e)
(1 − ξ(e))(|dρ − a| − dϕ )(e)
e∈supp− (dρ −a)
+
X
ω∈ET ,ϕ
πω {ξ(δXω \ W ) + (1 − ξ)(δXω ∩ W ) − h[dρ − a](Xω )},
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Primal integrality
where f (e) :=
nonnegative.
P
P ∈ΠG,S ,e∈P
f (P ) and W = supp− (dρ − a). By feasibility, all terms are
The third lemma, most essential, is about the stability of dual solutions.
Lemma 3.13. Let (ρ, ϕ, T ) be a bipartite feasible triple. If (ρ, ϕ, T ) is not optimal,
then there exists a bipartite feasible triple (ρ∗ , ϕ∗ , T ∗ ) such that
(1) τ (ρ∗ , ϕ∗ , T ∗ ) ≤ τ (ρ, ϕ, T ) − 2, and
∗
∗
(2) (dρ + dϕ )(x, y) ≤ (dρ + dϕ )(x, y) + 2 for x, y ∈ V .
Proof. Contract each segment ω with h[dρ − a](Xω ) = 0 so that h[dρ − a](Xω ) = 1
for each ω ∈ ET ,ϕ ; the resulting (ρ, ϕ, T ) may not be bipartite, but is feasible and
optimal, and dϕ + dρ does not increase. We are going prove to the following:
Claim 3.14. There exists a feasible triple (ρ̃, ϕ̃, T̃ ) (not necessarily bipartite) such
that
τ (ρ̃, ϕ̃, T̃ ) < τ (ρ, ϕ, T ),
|dρ̃ (x, y) − dρ (x, y)| ≤ 1 (x, y ∈ V ),
|dϕ̃ (x, y) − dϕ (x, y)| ≤ 1 (x, y ∈ V ).
This means that there is another feasible triple sufficiently close to (ρ, ϕ, T ) having
a smaller objective value. Suppose that this claim is true. According to the proof
of Theorem 3.4 together with Lemma 3.8, we can take a bipartite feasible triple
∗
∗
(ρ∗ , ϕ∗ , T ∗ ) so that τ (ρ∗ , ϕ∗ , T ∗ ) ≤ τ (ρ̃, ϕ̃, T̃ ) and (dρ +dϕ )(x, y) ≤ (dρ +dϕ )(x, y)+2
(for initial (ρ, ϕ)). By the evenness (Lemma 3.11), we have (1).
Now we prove Claim 3.14. We consider two cases: g(dρ ) < τ (ρ, ϕ, T ) and g(dρ ) =
τ (ρ, ϕ, T ); see (3.1) for the definition of g = g G .
Case 1: g(dρ ) < τ (ρ, ϕ, T ). Then π ϕ,T is feasible but nonoptimal to (3.4) with
d = dρ fixed. By Karzanov’s polynomial uncrossing process (Lemma 2.3), we can take
a laminar feasible solution π̃ such it has a smaller objective value and is sufficiently
close to the original laminar feasible solution π ϕ,T . Take a metric-tree T̃ with map
ϕ̃ : V → T̃ realizing π̃. Then (ρ, ϕ̃, T̃ ) is a required feasible triple.
Case 2: g(dρ ) = τ (ρ, ϕ, T ). In this case, dρ is non-optimal to a convex optimization
(3.2). By convexity, for a sufficiently small > 0, there is another metric d0 on V
feasible to (3.2) such that g(d0 ) < g(dρ ), and
(3.16)
|d0 (x, y) − dρ (x, y)| ≤ (x, y ∈ V ).
According to Claim 3.3, we can take a map ρ0 : V → Γ̄ so that ρ0 (s) = ps for s ∈ S
0
and dρ (x, y) ≤ d0 (x, y) for x, y ∈ V . Then
(3.17)
d(ρ0 (x), ρ(x)) ≤ (< 1/2) (x ∈ V ).
Indeed, by assumption (3.15), there is s ∈ S with d(ρ0 (x), ρ(x)) + d(ps , ρ(x)) =
d(ps , ρ0 (x)). Thus we have
0
d(ρ0 (x), ρ(x)) = dρ (s, x) − dρ (s, x)
0
= (d0 (s, x) − dρ (s, x)) + (dρ (s, x) − d0 (s, x)) ≤ ,
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Primal integrality
0
where the last inequality follows from (3.16) and dρ ≤ d0 .
Next we take a tree T 0 together with a map ϕ0 : V → T 0 so that triple (ρ0 , ϕ0 , T 0 )
0
is feasible with τ (ρ0 , ϕ0 , T 0 ) = g(dρ ) < τ (ρ, ϕ, T ). By (3.17) and the fact that dϕ is
integer-valued, we have
0
supp± (dρ − a) ⊆ supp± (dρ − a).
(3.18)
By (3.17) and |dρ − a|(e) ≥ 1 for e ∈ δ G Xω , ω ∈ ET ,ϕ , we have δ G Xω ∩ supp− (dρ − a) =
0
δ G Xω ∩ supp− (dρ − a) for ω ∈ ET ,ϕ . Hence we have
0
h[dρ − a](Xω ) = h[dρ − a](Xω ) (ω ∈ ET ,ϕ ).
(3.19)
For α ∈ [0, 1], define (dα , πα ) by
0
dα = (1 − α)dρ + αdρ ,
0
0
πα = (1 − α)π ϕ,T + απ ϕ ,T .
One can verify from (3.18) and (3.19) that (dα , πα ) is feasible to (2.6), and that
τ (dα , πα ) = (1 − α)τ (ρ, ϕ, T ) + ατ (ρ0 , ϕ0 , T 0 ).
Also define ρα : V → Γ̄ as
ρα (x) := the point p ∈ Γ̄ satisfying
d(p, ρ(x)) = (1 − α)d(ρ(x), ρ0 (x)),
d(p, ρ0 (x)) = αd(ρ(x), ρ0 (x)).
By (3.17), it holds
dα = dρα
(0 ≤ α ≤ 1).
Take a sufficiently small α > 0, let ρ̃ := ρα and π̄ := πα . Then (dρ̃ , π̄) is sufficiently
close to (dρ , π ϕ,T ). Here supp+ π̄ is not necessarily laminar. As in Lemma 2.3, apply
Karzanov’s polynomial uncrossing process to π̄. We get a laminar solution π̃ sufficiently close to π ϕ,T . Take metric-tree T̃ with map ϕ̃ : V → T̃ realizing π̃. Then
(ρ̃, ϕ̃, T̃ ) is a required one.
Proof of Proposition 3.10. Take an optimal solution (ξ, f ) in (2.2). Let e = xy ∈
E be an edge with ξ(e) > 0. Here we may assume (ρ, ϕ)(x) = (ρ, ϕ)(y) and a(e) = 0.
Indeed, replace e by a series of two edges xz and zy, set a(xz) = 0 and a(zy) = a(xy).
Then the problem (2.2) does not change; from any feasible solution (ξ, f ) in the
original instance we get a feasible solution in the new one with the same objective
value by setting ξ(xz) = ξ(xy) := ξ(e), and the converse is also possible by contracting
edge xz. In particular e is deletable in the original graph if and only if xz is deletable
in the new graph. Also we can extend (ρ, ϕ) by defining (ρ, ϕ)(z) = (ρ, ϕ)(x), which
is optimal to the new instance (since it is feasible and has the same objective value).
Suppose indirectly that e is not deletable. Here (ρ, ϕ, T ) is also feasible to G−e with
G−e
τ
(ρ, ϕ, T ) = τ G (ρ, ϕ, T ) by dρ (e) = dϕ (e) = 0 = a(e). So (ρ, ϕ, T ) is not optimal
to G−e. According to Lemma 3.13, we can take a bipartite triple (ρ∗ , ϕ∗ , T ∗ ) for G−e
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Primal integrality
∗
∗
with properties (1-2). In particular, dρ (x, y) + dϕ (x, y) ≤ 2. So there are four cases
∗
∗
(dρ (x, y), dϕ (x, y)) = (0, 0), (2, 0), (1, 1), (0, 2). The first case is clearly impossible
because (ρ∗ , ϕ∗ , T ∗ ) is also feasible to G with τ G (ρ∗ , ϕ∗ , T ∗ ) = τ G−e (ρ∗ , ϕ∗ , T ∗ ) <
∗
val(µ; G, a) ≤ τ G (ρ∗ , ϕ∗ , T ∗ ). In the following, we denote hG [dρ − a] simply by hG ;
we do not modify ρ∗ .
∗
∗
Case 1: (dρ (x, y), dϕ (x, y)) = (2, 0). In this case, triple (ρ∗ , ϕ∗ , T ∗ ) is also feasible
∗
∗
to G. Moreover (dρ −a)+ (E) = (dρ −a)+ (E\e)+2, and hG ◦(ϕ∗ , T ∗ ) = hG−e ◦(ϕ∗ , T ∗ )
since hG−e (Xω ) = hG (Xω ) for every ω ∈ ET ∗ ,ϕ∗ . Thus we have
τ G−e (ρ∗ , ϕ∗ , T ∗ ) + 2 ≤ val(µ; G, a) ≤ τ G (ρ∗ , ϕ∗ , T ∗ ) = τ G−e (ρ∗ , ϕ∗ , T ∗ ) + 2.
Therefore (ρ∗ , ϕ∗ , T ∗ ) is also optimal for G. In particular (ρ∗ , ϕ∗ , T ∗ ) fulfills the
∗
∗
optimality criterion (Lemma 3.12) with (ξ, f ). By (b), 0 = dϕ (e) < dρ (e) = 2
implies ξ(e) = 0, which contradicts ξ(e) > 0.
∗
∗
Case 2: (dρ (x, y), dϕ (x, y)) = (1, 1). Also in this case, triple (ρ∗ , ϕ∗ , T ∗ ) is feasible
∗
∗
to G. Obviously (dρ − a)+ (E) = (dρ − a)+ (E \ e) + 1. There is unique ω ∈ ET ∗ ,ϕ∗
such that πω = 1 and e ∈ δ G Xω . Therefore
hG ◦ (ϕ∗ , T ∗ ) = hG−e ◦ (ϕ∗ , T ∗ ) − hG−e (Xω ) + hG (Xω ).
Necessarily hG−e (Xω ) = 1 and hG (Xω ) = 0; otherwise τ G (ρ∗ , ϕ∗ , T ∗ ) < val(µ; G). So
(ρ∗ , ϕ∗ , T ∗ ) is also optimal to G. By πω > 0 and hG (Xω ) = 0, Lemma 3.12 (c) implies
∗
δXω · ξ − ξ(F ) + |F | − ξ(F ) = 0, where F := δ G Xω ∩ supp− (dρ − a). Then ξ(e0 ) = 0
for each e0 ∈ δ G X \ supp− (dρ − a). Thus ξ(e) = 0 since e ∈ δ G X ∩ supp+ (dρ − a).
Again this contradicts ξ(e) > 0.
∗
∗
Case 3: (dρ (x, y), dϕ (x, y)) = (0, 2). In this case, (ρ∗ , ϕ∗ , T ∗ ) is not feasible for G.
We modify it as follows. Let q := ϕ∗ (x), q 0 := ϕ∗ (y), and q̄ ∈ T be the unique point
with d(q, q̄) = d(q 0 , q̄) = 1. We may assume q̄ ≺ q. Also we may assume ϕ−1 (q̄) 6= ∅
(by adding dummy isolate nodes). Then open segments ω := (q̄, q), ω 0 := (q̄, q 0 )
(of unit length) belong to ET̃ ∗ ,ϕ̃∗ . Let X := Xω and let Y := Xω0 . Delete open
segment ω = (q̄, q) from T ∗ , and then identify q and q 0 . Let (T̃ ∗ , ϕ̃∗ ) be the resulting
metric-tree with a map. Then (ρ∗ , ϕ̃∗ , T̃ ∗ ) is bipartite and feasible to G. Obviously
∗
∗
(dρ − a)+ (E) = (dρ − a)+ (E \ e).
Suppose q 0 ≺ q̄. Then
hG ◦ (ϕ̃∗ , T̃ ∗ ) − hG−e ◦ (ϕ∗ , T ∗ ) = −hG−e (X) − hG−e (Y ) + hG (X ∪ Y ).
Necessarily hG−e (X) = hG−e (Y ) = 1 and hG (X ∪ Y ) = hG−e (X ∪ Y ) = 0 (by (2.7)).
Therefore (ρ∗ , ϕ̃∗ , T̃ ∗ ) is optimal to G. Consider edge e0 ∈ δ G X \ {e}. If e0 joins X
∗
∗
∗
and Y , then dϕ̃ (e0 ) < dϕ (e0 ) ≤ |dρ − a|(e). Otherwise e0 ∈ δ G (X ∪ Y ). Thus, by
optimality criterion (Lemma 3.12 (b,c)) for hG (X ∪ Y ) = 0, we have
∗
1 if e0 ∈ supp− (dρ − a),
0
(3.20)
ξ(e ) =
(e0 ∈ δ G X \ {e}).
0 otherwise,
Decompose ξ into a convex combination of the incidence vectors of inner-odd-joins
F1 , F2 , . . . , Fk . Since ξ(e) > 0, then there is an index j such that e ∈ Fj . By (3.20),
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Section 4. Polynomial solvability and NP-hardness
∗
∗
Fj ∩ δ G X = {e} ∪ (δ G X ∩ supp− (dρ − a)). Now |δ G X \ supp− (dρ − a)| is even, since
hG−e (X) = 1. Thus we have
∗
|Fj ∩ δ G X| = 1 + |δ G X ∩ supp− (dρ − a)|
∗
= 1 + |δ G X| − |δ G X \ supp− (dρ − a)|
≡ 1 + |δ G X| mod 2.
This is a contradiction since for any inner-odd-join F and X ⊆ V \ S, |F ∩ δ G X| and
|δ G X| must have the same parity. The argument for the case q̄ ≺ q 0 is essential same;
replace X ∪ Y by Y \ X in the argument above. Now the proof of Proposition 3.10 is
complete.
4
Polynomial solvability and NP-hardness
In this section, we prove the polynomial time solvability of µ-CEDP for a tree metric
µ, and the NP-hardness of µ-EDP when µ is not a truncated tree metric.
4.1
Polynomial time solvability
As in [27], our approach is based on the polynomial equivalence between separation
and optimization [12], We are given an instance G = (V, E), S, a, µ of µ-CEDP. Suppose that µ is a tree metric. The linear programming (2.2) has an exponential number
of variables and inequalities. We can reduce the number of variables to a polynomial
size by using the edge-node formulation of multiflows, instead of the edge-path formulation f : ΠG,S → R+ . Also the separation of the inner-odd-join polytope QG,S can
be done in polynomial time by any T -join algorithm together with the equivalence
of optimization and separation [12]. Thus again by the ellipsoid method [12] we can
evaluate val(µ; G, a) in polynomial time. Consequently, for an edge e ∈ E, we can
determine whether e is deletable or not. If e is deletable, then delete it. Repeat it
at most |E| times until all edges are not deletable; once e becomes undeletable then
e keeps undeletable in the subsequent steps. Then the resulting graph G is inner
Eulerian by Proposition 3.10. By any algorithm for the laminar locking theorem, e.g.,
[20], we can find an edge-disjoint set of S-paths attaining val(µ; G, 0) in polynomial
time. This is our desired solution for µ-CEDP.
A generalization to the capacitated setting is straightforward. Now we are also
given
an integer-valued edge-capacity c : E → Z+ . The problem is to maximize
P
flow-value functions f : ΠG,S →
P ∈ΠG,S f (P )(µ(sP , tP ) − a(P )) over integer-valued
P
Z+ satisfying the capacity constraint
P ∈ΠG,S ,e∈P f (P ) ≤ c(e) for e ∈ E. Let
val(µ; G, c, a) denote the maximum value. The LP-relaxation can be constructed
as follows. Let QG,c,S be the convex hull of all integer-valued function ξ : E → Z+
G
such that
P 0 ≤ ξ(e) ≤ c(e) for e ∈ E andPξ(δ x) is even for x ∈ V \ S. Then replace
ξ(e) + P ∈ΠG,S ,e∈P f (P ) ≤ 1 by ξ(e) + P ∈ΠG,S ,e∈P ≤ c(e), and replace ξ ∈ QG,S by
ξ ∈ QG,c,S in (2.2). The resulting LP has also an integer optimal solution if µ is a tree
metric. Since QG,c,S is a parity-constraint capacitated b-matching polytope, the linear
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NP-hardness
optimization for QG,c,S is polynomially solvable, and then the separation for QG,c,S
can also be done polynomial time. Thus we can evaluate val(µ; G, c, a) in polynomial
time. Consequently, for each edge e ∈ E we can determine the maximum deletion
αc (e) := max{α ∈ [0, c(e)] ∩ Z | val(µ; G, c − αχe , a) = val(µ; G, c, a)} in polynomial
time by the bisection method, where χe is the incidence vector of e. The rest is the
same as above.
4.2
NP-hardness
The goal of this subsection is to prove the NP-hardness of µ-EDP when µ is not a
truncated tree metric (Theorem 1.3). We will prove two auxiliary results. The first
one is a 4-point characterization of truncated tree metrics.
Theorem 4.1. A weight µ is a truncated tree metric if and only if for all sets of four
distinct elements s, t, u, v ∈ S, at least one of the following inequalities holds:
(4.1)
(4.2)
(4.3)
(4.4)
µ(s, t) = 0,
µ(u, v) = 0,
µ(s, t) + µ(u, v) ≤ µ(s, v) + µ(t, u),
µ(s, t) + µ(u, v) ≤ µ(s, u) + µ(t, v).
Note that here we do not require the triangle inequality to hold. Therefore, if
µ is not a truncated tree metric, then there are four elements s, t, u, v such that
all inequalities (4.1)-(4.4) do not hold. From this violating quadruple {s, t, u, v},
we will reduce some NP-hard problem to µ|{s,t,u,v} -EDP. For the reduction, we use
the following version of the integer 2-commodity flow problem for which one of the
commodities is unit:
Problem 4.2. The (1, ∗)-edge-disjoint paths problem takes the input of an undirected
graph G = (V, E), nodes r, q ∈ V , and disjoint sets R, Q ⊆ V such that {r}, {q}, R, Q
are disjoint and |R| = |Q|. The goal is to determine paths P0 , P1 , · · · , P|R| such that
they are pairwise edge-disjoint, and P0 connects r with q, and Pi connects a node in
R with a node in Q, for i = 1, · · · , |R|, or establish that there is no such a family of
paths.
Even, Itai, and Shamir [10] actually proved that a directed version of (1, ∗)-edgedisjoint paths problem is NP-hard. Their reduction works for our undirected version
with small change.
Theorem 4.3 (essentially [10]). The (1, ∗)-edge-disjoint paths problem is NP-hard.
Now we prove Theorem 1.3 assuming Theorems 4.1 and 4.3, which will be proved
later.
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NP-hardness
Proof of Theorem 1.3. Suppose that µ is not a truncated tree metric. Then
Theorem 4.1 implies that there are four distinct elements s, t, u, v such that each of
the following inequalities holds:
(4.5)
(4.6)
(4.7)
(4.8)
µ(s, t) > 0,
µ(u, v) > 0,
µ(s, t) + µ(u, v) > µ(s, v) + µ(t, u),
µ(s, t) + µ(u, v) > µ(s, u) + µ(t, v).
Then either both of
(4.9)
(4.10)
2µ(s, t) + 2µ(u, v) > µ(s, u) + µ(s, v) + µ(u, v),
2µ(s, t) + 2µ(u, v) > µ(t, u) + µ(t, v) + µ(u, v),
or both of
(4.11)
(4.12)
2µ(s, t) + 2µ(u, v) > µ(s, t) + µ(s, v) + µ(t, v),
2µ(s, t) + 2µ(u, v) > µ(s, t) + µ(s, u) + µ(t, u)
hold. Indeed, assume that at most one of (4.9)-(4.10), and most at one of (4.11)-(4.12)
holds. By symmetry, we may assume that (4.9) and (4.11) do not hold. This implies
that
3µ(s, t) + 3µ(u, v) ≤ µ(s, u) + 2µ(s, v) + µ(t, v).
On the other hand, (4.5)-(4.8) imply that
3µ(s, t) + 3µ(u, v) > 2µ(s, v) + 2µ(t, u) + µ(s, u) + µ(t, v) ≥
≥ µ(s, u) + 2µ(s, v) + µ(t, v).
This is a contradiction. So, by symmetry, we may assume that each of the inequalities
(4.9)-(4.10) hold.
Now consider an instance of the (1, ∗)-edge-disjoint paths problem, that is an undirected graph G = (V, E), nodes r, q ∈ V , and disjoint sets R, Q ⊆ V , such that
{r}, {q}, R, Q are disjoint and |R| = |Q|. Define G0 := (V 0 , E 0 ) by adding 4 new nodes
to G, namely s, t, u, v, and add new edges ur and vq, and sx for all x ∈ R, ty for all
y ∈ Q. A solution to the µ|S 0 -EDP with G0 , S 0 := {s, t, u, v} defines a set of paths so
that each path has both its endpoints in S 0 . If this set of paths contains exactly 1 path
from u to v, and |R| paths from R to Q, then the weight is equal to |R|µ(s, t)+µ(u, v).
We claim that this routing – i.e. having 1 path from u to v, and |R| paths from s to t
– is the only way to achieve a weight of at least |R|µ(s, t) + µ(u, v). This follows from
the following claim.
Claim 4.4. Consider the perfect graph KS 0 with node set S 0 := {s, t, u, v}, and edge
weights given by µ, and node-capacity b(s) = b(t) = |R| and b(u) = b(v) = 1. Then
ξ(st) = |R| and ξ(uv) = 1 and ξ(e) = 0 for other edge e gives the unique maximum
integral b-matching in KS 0 .
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NP-hardness
Proof. Recall that an integral b-matching is a vector ξ : EKS 0 → Z+ such that
ξ · δ{w} ≤ b(w) for all w ∈ S 0 . Let ξ be a maximum weight integral b-matching with
respect to weight µ|s,t,u,v . First suppose that ξ(su) ≥ 1 and ξ(tv) ≥ 1. Then, by (4.8),
decreasing ξ on su and tv by one, and increasing x on st and uv by one will increase
the weight of ξ. This is a contradiction. This implies that ξ(su) = 0 or ξ(tv) = 0. By
symmetry we may assume ξ(su) = 0. Similarly we have (i) ξ(sv) = 0 or (ii) ξ(ut) = 0.
Case (i): ξ(sv) = 0. Now suppose that ξ(tu) ≥ 1 and ξ(tv) ≥ 1. Then decrease
ξ on tu and tv by one, increase ξ on st by two, and increase ξ on uv by one. This,
by (4.10), implies that the weight of ξ increases, a contradiction. Thus we get that
ξ(tu) = 0 or ξ(tv) = 0. Say, by symmetry, ξ(tu) = 0. To sum up, we now have
ξ(su) = ξ(sv) = x(tu) = 0. If ξ(tv) ≥ 1, then decrease ξ on tv by one, and increase ξ
on st and uv by one, to experience the weight of ξ increased by (4.7). A contradiction,
showing that ξ(su) = ξ(sv) = ξ(tu) = ξ(tv) = 0. By (4.5) and (4.6), and by the
assumption that ξ has maximum weight, we get that ξ(st) = |R| and ξ(uv) = 1.
Case (ii): ξ(ut) = 0. Since b(v) = 1, at most one of ξ(sv) and ξ(tv) is positive. By
the same argument, we get ξ(sv) = ξ(tv) = 0, and hence ξ(st) = |R| and ξ(uv) = 1,
as required. This completes the proof.
Thus we conclude that there is a polynomial reduction from the (1, ∗)-edge-disjoint
paths problem to µ-EDP, implying that µ-EDP is NP-hard, proving Theorem 1.3.
The reminder of this subsection is devoted to proving Theorems 4.1 and 4.3.
Characterization of truncated tree metrics. Here we prove Theorem 4.1. We
start with a well-known 4-point characterization of tree metrics; see [40].
Theorem 4.5. A weight µ is a tree metric if and only if it satisfies the triangle
inequality, and for all sets of four distinct elements s, t, u, v ∈ S, at least one of the
following inequalities holds:
(4.13)
(4.14)
µ(s, t) + µ(u, v) ≤ µ(s, v) + µ(t, u),
µ(s, t) + µ(u, v) ≤ µ(s, u) + µ(t, v).
Hirai [14] considered the distance among subtrees in a tree, and derived a similar
4-point characterization for it. A weight µ is called a subtree distance if there are a
metric-tree T and a family {Rs }s∈S of its subtrees (connected subsets) indexed by S
such that
µ(s, t) = dT (Rs , Rt ) (s, t ∈ S).
In this case we say that (T , {Rs }s∈S ) realizes µ or is a realization of µ. A truncated
tree metric is exactly a subtree distance so that each subtree is a ball around some
point. Hirai provided the following characterization of subtree distances.
Theorem 4.6 ([14]). A weight µ is a subtree distance if and only if for all sets of
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four distinct elements s, t, u, v ∈ S, at least one of the following inequalities holds:
(4.15)
(4.16)
(4.17)
(4.18)
(4.19)
(4.20)
(4.21)
(4.22)
µ(s, t) = 0,
µ(u, v) = 0,
µ(s, t) + µ(u, v) ≤ µ(s, v) + µ(t, u),
µ(s, t) + µ(u, v) ≤ µ(s, u) + µ(t, v),
2µ(s, t) + 2µ(u, v) ≤ µ(s, u) + µ(s, v) + µ(u, v),
2µ(s, t) + 2µ(u, v) ≤ µ(t, u) + µ(t, v) + µ(u, v),
2µ(s, t) + 2µ(u, v) ≤ µ(s, t) + µ(s, v) + µ(t, v),
2µ(s, t) + 2µ(u, v) ≤ µ(s, t) + µ(s, u) + µ(t, u).
Note that if a weight function satisfies (4.1)-(4.4), it will clearly also satisfy (4.15)(4.22).
We are ready to prove Theorem 4.1. First we prove the “only if” part. Consider a
realization (T , {Rs }s∈S ) of the truncated tree metric as a subtree distance. Now each
subtree Rs is a ball B(ps , rs ) with center ps and radius rs . Let µ̄(s, t) := dT (ps , pt ) for
s, t ∈ S. Then µ̄ is a tree metric. Take any quadruple s, t, u, v ∈ S. We may assume
that µ(s, t) > 0 and µ(u, v) > 0, for otherwise (4.1) or (4.2) holds. That implies
µ(s, t) = µ̄(s, t) − rs − rt and µ(u, v) = µ̄(u, v) − ru − rv . By Lemma 4.5, at least one
of (4.13) and (4.14) holds for µ̄. Say, µ̄(s, t) + µ̄(u, v) ≤ µ̄(s, u) + µ̄(t, v), for example.
Then µ(s, t) + µ(u, v) = µ̄(s, t) + µ̄(u, v) − rs − rt − ru − rv ≤ µ̄(s, u) + µ̄(t, v) − rs −
rt − ru − rv ≤ µ(s, u) + µ(t, v). This implies (4.4).
Second, using Lemma 4.6, we prove the “if” part. By Lemma 4.6, if µ satisfies at
least one of (4.1)-(4.4) for every s, t, u, v ∈ S, then it must be a subtree distance.
Now suppose that µ is realized by (T , {Rs }s∈S ). Moreover we may assume that every
leaf p ∈ T of the tree appears as a single-node subtree Rs = {p} (for some s ∈ S) –
otherwise we could chop off part of T , and retain a realization of µ. Then we claim
that every subtree Rs actually is a ball, which will prove the Lemma. For a fixed s ∈ S,
if the diameter of Rs is zero, then we are done. Otherwise consider the median of Rs ,
that is a point x ∈ Rs such that there are two distinct points x1 , x2 in Rs at maximum
distance from x. Let r := dT (x, x1 ) = dT (x, x2 ) = maxx0 ∈Rs dT (x, x0 ). We can take
leaves y1 , y2 of T such that dT (y1 , y2 ) = dT (y1 , Rs ) + dT (x1 , x2 ) + dT (Rt , y2 ). We claim
that Rs = B(x, r). If by contradiction this were not true, then there would be a leaf y3
such that dT (Rs , y3 ) > dT (B(x, r), y3 ). Clearly y3 6= y1 , y2 . By our assumption, there
must be s1 , s2 , s3 such that Rs1 = {y1 }, Rs2 = {y2 }, Rs3 = {y3 }. Clearly, s, s1 , s2 , s3
are four distinct elements of S. We claim that for s = s, t = s3 , u = s2 , v = s1 , none
of the inequalities (4.1)-(4.4) holds: To check this, note that y1 6= y2 , which proves
that (4.2) does not hold. From dT (Rs , y3 ) > dT (B(x, r), y3 ) we get that y3 ∈
/ Rs ,
which proves that (4.1) does not hold. We also see that µ(s, t) + µ(u, v) = dT (y1 , y2 ) +
dT (Rs , y3 ) = dT (y1 , x)+dT (x, y2 )+dT (Rs , y3 ) > dT (y1 , x)+dT (x, y2 )+dT (x, y3 )−r ≥
dT (y1 , y3 ) + dT (x, y2 ) − r = dT (y1 , y3 ) + dT (Rs , y2 ) = µ(s, u) + µ(t, v), that is, (4.4)
does not hold. A similar computation show that (4.3) does not hold either. The
contradiction proves that Rs is a ball for all s ∈ S, completing the proof of the
Lemma. Based on the algorithm from Hirai [14], and the above proof, we get the
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30
following.
Corollary 4.7. There is a polynomial time algorithm to decide whether a given weight
µ is a truncated tree metric, and if so, also provide a realization (T , {B(ps , rs )}s∈S ).
Hardness of (1, ∗)-edge-disjoint paths problems. The proof of Theorem 4.3 is
a slight modification of [10]. Consider an instance of 3-SAT with variables x1 , · · · , xn ,
and clauses c1 , · · · , cm . Each clause cj has three literals like xi or xi .
Now we construct an instance of the (1, ∗)-edge-disjoint paths problem. We introduce nodes r, q. For all i = 1, · · · , n, we introduce nodes si , ti . For all i = 1, · · · , n and
i
i
, q2j
. We introduce edges to create
j = 1, · · · , m, we introduce nodes pi2j−1 , pi2j , q2j−1
i
i
i
i
paths si , p1 , p2 , · · · , p2m , ti , and edges to create paths si , q1i , q2i , · · · , q2m
, ti . We add
j
edges rs1 , ti si+1 , tn q. For all j = 1, · · · , m, we add nodes a , bj , and for k = 1, 2, 3,
we add nodes bj,k , aj,k , aj,k , bj,k and edges aj bj,k , bj,k aj,k , bj bj,k , bj,k aj,k . For a literal
k
cj = xu1 ∨ xu2 ∨ xu3 , and for k = 1, 2, 3, we add edges aj,k pu2j−1
and bj,k pu2jk . If instead
uk
uk
of xuk , we have xuk in literal cj , then, instead, we add edges aj,k q2j−1
and bj,k q2j
. Let
0
G = (V, E) denote the graph constructed this way. Let R be the set of all nodes
aj , aj,k , aj,k , and let Q0 be the set of all nodes bj , bj,k , bj,k . Let R and Q be sets introducing a copy of a node in R0 and Q0 , respectively, and connect each of the nodes in
R0 and Q0 with its copy.
The purpose of the construction is the following claim: the 3-SAT formula is satisfiable if and only if there is a set of edge-disjoint paths such that one of the paths goes
from r to q, and the remaining paths are connecting a node in R with a node in Q. For
the if part, consider a satisfying assignment. Then we use a path from s to t traversing
all nodes si , ti in this order, using the paths qli if xi was “true”, and using the paths
pil if xi was “false”. For each j, clause cj has a true literal xuk = “true” or xuk =
uk
uk j,k
k
“true”. Use path aj,k , pu2j−1
, pu2jk , bj,k if xi = “true”, and use path aj,k , q2j−1
, q2j
, b if
j,k
j,k
j
xi =“true”. So we get a path connecting a and b . Furthermore a is matched to
0
bj,k and bj is matched aj,k by one edge. Also for k 0 ∈ {1, 2, 3} \ k, bj,k is matched to
0
aj,k , and bj,k0 is matched to aj,k0 . Finally add a perfect matching between R, Q and
R0 , Q0 . We get a solution of the instance of the (1, ∗)-edge-disjoint paths problem.
Conversely, suppose there is a solution to the (1, ∗)-edge-disjoint paths problem. This
contains one path P0 from r to q, and a set of paths connecting each node in R with a
node in Q. This path must go via the paths pil or the paths qli , for otherwise one of the
terminals in R or Q would be cut off. We set xi = “true” if P0 use qli , and xi = “false”
if P0 use qli . We show that this assignment is satisfiable. It is also straightforward
that for each j, the copies of nodes aj , bj , bj,k , aj,k , aj,k , bj,k (k = 1, 2, 3) are paired with
each other, except for one of the nodes aj,k , and one of the nodes bj,k . In particular
uk
the unpaired node aj,k is necessarily incident to pu2jk or q2j
not used by P0 . This means
that clause cj has a true literal in this assignment for each j.
5
Concluding remarks
Here we discuss several related issues and raise open problems.
EGRES Technical Report No. 2010-13
Section 5. Concluding remarks
31
Node-disjoint version. As mentioned, Mader [32] extended (1.1) to a node-disjoint
version. One may naturally ask a node-disjoint generalization of our results. Nodedisjoint problems have a quite different nature from edge-disjoint problems. Indeed,
an LP-based approach to Mader’s node-disjoint S-paths theorem is not known. Also
there are algebraic generalizations to group-labeled graphs [8, 33].
Recently Pap [35] proved a min-max theorem and the polynomial solvability of
node-disjoint S-paths packing for truncated tree metrics. His approach is quite different from that presented in this paper, and its relation is not clear. Also we do not
know whether his approach can be extended to a cost version and has a polyhedral
interpretation.
Combinatorial polynomial time algorithm. Our algorithm crucially depends
on the ellipsoid method. Also Karzanov’s algorithm [23] for mincost edge-disjoint Spaths is lengthy and much complicated, though it is combinatorial. So it is desirable to
find a modest combinatorial polynomial time algorithm. Such algorithms for disjoint
S-paths packing are (i) (a reduction to) Lovasz’s linear matroid matching [29, 39],
(ii) Chudnovsky-Cunningham-Geelen’s algorithm for non-zero A-paths [7], and (iii)
Pap’s algorithm for non-returning A-paths [34]. They solve node-disjoint problems
(on group-labeled graphs for (ii) and (iii)). A natural approach is to modify or extend
them (i-iii) to the cost and tree-metric version.
We here suggest an alternative approach. As mentioned in the introduction, our
formula includes a discrete minimax relation for inner-odd-joins F and tree locations
ρ:
1 ρ
1 ρ
dΓ − a (E \ F ) = min max
dΓ − a (E \ F ).
(5.1)
max min
ρ
ρ
F
F
2
2
Here the LHS coincides with val(µ; G, a) by (2.1). So our problem also reduces to
finding a saddle point in (5.1). Once a saddle point (F, ρ) is found, an optimal Spaths packing is obtained by solving an easy multiflow problem on an Eulerian graph
G − F , according to Lemma 2.1. Such a game-theoretic consideration may bring a
new algorithm to this problem.
Integrality gap of the LP-relaxation. The LP-relaxation 2.2 is applicable to
µ-EDP/CEDP for any weight µ. Many existing approximation algorithms
for µ-EDP
p
use a natural LP-relaxation [4, 13, 42], which is known to have Ω( |V |)-integrality
gap. Our LP-relaxation (2.2) is stronger than the natural LP. So it would be interesting to estimate the integrality gap of (2.2) and its dependency of µ, and to
design approximation algorithms based on it. For example, consider the integer 2commodity flow maximization, corresponding to µ = µ2com defined as S = {s, t, s0 , t0 },
µ2com (s, t) = µ2com (s0 , t0 ) = 1, and the other weights are zero. By Rothschild-Winston
theorem [37], for every inner Eulerian graph, the natural multiflow relaxation has
an integral optimal solution with the value equal to the minimum 2-commodity cut.
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Section 5. Concluding remarks
Thus the inequality (generally strict) holds:
val(µ2com ; G, 0) = max val(µ2com ; G − F, 0) = max min |δ G−F X|
F
≤
F
X
G
max min(1 − ξ)(δ X) ≤ min max |δ G−F X|,
ξ∈QG,S
X
X
F
where F denotes an inner-odd-join and X denotes a vertex subset with s, s0 ∈ X 63 t, t0
or s, t0 ∈ X 63 t, s0 . Based it, can we design a better approximation algorithm than
the trivial 2-approximation algorithm ?
Relation to Karzanov’s approach. In [23], Karzanov established a min-max relation for µ-CEDP with µ = p1 (p > 0); see also [25]. His min-max formula takes
a form different from our min-max formula (1.2) or Theorem 3.4. Here we discuss
their relationship. We need some notions to explain Karzanov’s min-max formula. An
inner fragment φ is a pair (Xφ , Uφ ) of Xφ ⊆ V \ S and Uφ ⊆ δ G Xφ with |Uφ | odd. Let
F0 denote the set of all inner fragments. The characteristic function χφ of an inner
fragment φ is a function on E defined by

if e ∈ Uφ ,
 1
−1 if e ∈ δ G Xφ \ Uφ ,
χφ (e) =
(e ∈ E).

0
otherwise,
Given β : F 0 → R+ and γ : E → R+ , define the function lβ,γ on E by
X
lβ,γ := a + γ +
β(φ)χφ .
φ∈F 0
We say that (β, γ) is p-admissible if lβ,γ ≥ 0 and lβ,γ (P ) ≥ p for all P ∈ ΠG,S .
Theorem 5.1 ([23]). For any p ≥ 0, we have
(5.2)
val(p1; G, a) = min γ(E) +
X
φ∈F 0
βφ (|Uφ | − 1),
where the minimum is taken over all p-admissible pairs.
In fact, the minimization problem in RHS of (5.2) is the LP-dual of the LP obtained
from (2.2) by substituting (2.4) to it and eliminating variable ξ. Then Theorem 5.1
follows from Theorem 3.9. Instead of detailed calculations, we explain how to obtain
(β, γ) from our (ρ, ϕ, T ), and answer affirmatively Karzanov’s question [25, Section
6, (2)]: does there exist a half-integral optimal p-admissible pair (β, γ) when p and a
are both integral ?
Suppose that p is even, and a is even-valued. For each s ∈ S, consider a path
Ps with length p/2 and ends qs , ps . Next identify qs over all s ∈ S. The resulting
tree (a subdivision of star) is denoted by Γ . Then (Γ, {ps }s∈S ; 1) realizes p1. By
Theorem 3.4, we can take an optimal feasible triple (ρ, ϕ, T ) such that T = T̄ for
EGRES Technical Report No. 2010-13
33
References
some tree T , ρ(V ) ⊆ V Γ , ϕ(V ) ⊆ V T , and hG [dρ − a](Xω ) = 1 for ω ∈ ET ,ϕ . Define
(β, γ) by
πω if ∃ω ∈ ET ,ϕ : Xφ = Xω , Uφ = δ G Xω \ supp− (dρ − a),
β(φ) :=
(φ ∈ F 0 ),
0 otherwise,
ρ
(d − a − dϕ )(e) if e 6∈ supp− (dρ − a),
γ(e) :=
(e ∈ E).
0
if e ∈ supp− (dρ − a),
Then lβ,γ is given by
dρ (e)
if e 6∈ supp− (dρ − a),
(e ∈ E),
(a − dϕ )(e) if e ∈ supp− (dρ − a),
P
P
where we use dϕ (e) = − φ β(φ)χφ (e) if e ∈ supp− (dρ −a) and dϕ (e) = φ β(φ)χφ (e)
if e 6∈ supp− (dρ − a). In particular lβ,γ (e) ≥ dρ (e) ≥ 0 (by |dρ − a|(e) ≥ dϕ (e)). Hence
lβ,γ (P ) ≥ dρ (P ) = p for P ∈ ΠG,S , implying that (β, γ) is admissible. Moreover,
X
γ(E) +
βφ (|Uφ | − 1)
l
β,γ
(e) =
φ∈F 0
+
= (d − a) (E) − dϕ (E \ supp− dρ − a)
X
πω {δXω (E \ supp− dρ − a) − hG [dρ − a](Xω )}
+
ρ
ω∈ET ,ϕ
= (dρ − a)+ (E) − hG [dρ − a] ◦ (ϕ, T ) = τ G (ρ, ϕ, T ).
Thus (β, γ) is optimal, and also integral by construction.
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