Multicommodity flow, well-linked
terminals and routing problems
Chandra Chekuri
Lucent Bell Labs
Joint work with Sanjeev Khanna and Bruce Shepherd
Mostly based on paper in STOC ‘05
Routing Problems
Input: Graph G(V,E), node pairs s1t1, s2t2, ..., sktk
Goal: Route a maximum # of si-ti pairs
Route?
EDP: path for each pair, paths edge disjoint
NDP: paths are node disjoint
AN-Flow: flow of one unit per pair with edge/node
capacity equal to 1
Disjoint paths vs An-Flow
s1
s2
s1
1/2
s2
1/2
t2
t1
t2
1/2
1/2
t1
Setup
Terminals: X = {s1,t1,s2,t2,...,sk,tk}
each terminal occurs in exactly one pair, |X| = 2k
Pairs: matching M on X
Instance: (G,X,M)
unit capacity graph
Focus: edge problems, EDP and An-flow.
Multicommodity Flow Formulation (IP)
P(i) : set of paths between si and ti
P = P(1) [ P(2) ... [ P(k)
f(p) : 1 if flow on path p 2 P, 0 otherwise
xi : 1 if siti is routed, 0 otherwise
max i xi s.t
xi = p 2 P(i) f(p)
p: e 2 p f(p) · 1
xi, f(p) 2 {0,1}
1·i·k
e2E
Multicommodity Flow Formulation (LP)
P(i) : set of paths between si and ti
P = P(1) [ P(2) ... [ P(k)
f(p) : flow on path p 2 P
xi : amount of flow routed for siti
max i xi s.t
xi = p 2 P(i) f(p)
p: e 2 p f(p) · 1
xi, f(p) 2 [0,1]
1·i·k
e2E
Framework
1. Start with an LP solution.
2. Use LP solution to decompose the input
instance into a collection well-linked
instances.
3. Use well-linkedness to route large fraction
Outline
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
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Cut vs Flow well-linkedness
Well-linked decomposition
Multicommodity flow to well-linked decomp
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decomposition via cuts
fractional well-linkedness to well-linkedness
Conclusions
Multicommodity Flows
MC Flow instance:
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capacitated graph G
non-negative demand matrix d on V x V
route dij flow for node pair ij
Product MC Flow instance:


node weights p : V ! R+
implicitly defines d with dij = p(i)p(j) / p(V)
Sparse Cuts and Multicomm. Flow
Given a cut (S, V-S) in G and demand matrix d:
sparsity of S = |d(S)| / d(S,V-S)
S
V-S
MCflow for d is feasible in G implies sparsity ¸ 1
Sparse Cuts and MC Flow
MCflow for d is feasible in G implies sparsity ¸ 1
d is feasible in G if sparsity = W(log n)
[LR88,LLR94,AR94]
For product MC Flow in planar G, sparisty of W(1)
sufficient [KPR93]
Flow-cut gap b(G): minimum sparsity reqd for
guaranteeing mc flow
Cut-Well-linked Set
Subset X is cut-well-linked in G if for every
partition (S,V-S) , # of edges cut is at least # of
X vertices in smaller side
S
V-S
for all S ½ V with |S Å X| · |X|/2, |d(S)| ¸ |S Å X|
Flow-Well-linked Set
Subset X is flow-well-linked in G if the following
multicommodity flow is feasible in G:
for u,v in X, d(uv) = 1/|X|
product (uniform) multicommodity flow on X
p(u) = 1 if u 2 X
= 0 otherwise
Cut vs Flow well-linked
X flow-linked ) X is ~cut-linked
X cut-linked ) X flow-linked with congestion b(G)
b(G) – worst case flow-cut gap for product
multicommodity instances in G
Weighted versions
p: X ! R+ weight function on X
p(v) : weight of v in X
p-cut-linked: for all S ½ V with p(S Å X) · p(X)/2,
|d(S)| ¸ p(S Å X)
p-flow-linked: multicommodity flow instance with
d(uv) = p(u) p(v) / p(X) is feasible in G
Well-linked instance of EDP
Input instance: G, X, M
X = {s1, t1, s2, t2, ..., sk, tk} – terminal set
M : matching on X
(s1,t1), (s2,t2) ... (sk,tk)
X is well-linked in G
Well-linked instance: weighted
Input instance: G, X, M
X = {s1, t1, s2, t2, ..., sk, tk} – terminal set
M : matching on X
(s1,t1), (s2,t2) ... (sk,tk)
X is p-well-linked in G for some p: X ! R+
Assume: p(v) · 1
Examples
s1
s2
t1
t2
s3
s4
t3
t4
s1
s2
t1
t2
s3
s4
t3
t4
Not a well-linked instance
A well-linked instance
Well-linked Decomposition
G, X, M
edge disjoint subgraphs
G1, X1, M1
Mi ½ M
G2, X2, M2
Xi is well-linked in Gi
Gr, Xr, Mr
i |Xi| ¸ OPT/a
Example
s1
s2
t1
t2
s3
s4
t3
t4
s1
t1
s2
t2
s3
t3
s4
t4
Well-linked Decomposition via Flow
G, X, M
Flow f
G1, X1, M1
G2, X2, M2
Xi is pi-flow-well-linked in Gi
Gr, Xr, Mr
i pi(Xi) ¸ f/a
Decomposition via trees/Racke
Simple decomposition for trees: a = O(1)
Represent G as a tree (approximately) [Racke03]
Done in [CKS04]
Decomposition based on recursive cuts [CKS05]
simple
better ratio
applies to node problems
Trees
Define p: X ! R+
p(sj) = p(tj) = fj the flow in LP
Suppose X is p/10-flow-well-linked done!
Otherwise exists cut of sparsity less than 1/10
Pick sparse cut (S,V-S) with S minimal
Trees
ce < p(S)/10
V-S
S
terminals in S are
p-well-linked!
Decomposition using Sparse Cuts
Start with LP soln for given instance
fj flow for pair sjtj : assume flow decomposition
f = j fj total flow in LP
define p: X ! R+
p(sj) = p(tj) = fj
Decomposition Algorithm
If X is p / 10 b(G) log k-flow-linked STOP
Else
Find a (approx) sparse cut (S,V-S) wrt p in G
Remove flow on edges of dG(S)
G1 = G[S], G2 = G[V-S]
Recurse on G1, G2 with remaining flow
Analysis
Remaining graphs at end of recursion
(G1,X1,p1) , (G2,X2,p2) , ...., (Gh, Xh, ph)
pi is the remaining flow for Xi
Xi is pi /10 b(G) log k flow-linked in G_i
i pi(Xi) ¸ Original flow - # edges cut
Bounding the number of edges cut
X is not p / 10 b(G) log k flow-linked
) |dG(S)| · p(S) / 10 log k
S
V-S
Analysis cont
Theorem: total number of edge cut is · f/2
T(x): max # of edges cut if started with flow x
T(f) · T(f1) + T(f2) + f1 / 10 log k
For f · k, T(f) · f/2
Analysis contd
i pi (Xi) ¸ f/2
Xi is pi/10 b(G) log k flow-well-linked
Fractional to integer well-linked
Theorem:
G, X, M input instance. X is p-flow-well-linked.
Then G, X’, M’ s.t
 M’ ½ M,


X’ is flow-well-linked
|X’| = W(p(X))
Edge case: spanning tree clustering
T spanning tree of G, rooted at r
Tv : subtree rooted at v
Can assume maximum degree of T is 4
1. Find deepest node u s.t p(Tu) ¸ 1
Note: p(Tu) · 5
2. Remove Tu from T
3. Continue until p(T) · 1
Spanning tree clustering
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Spanning tree clustering
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0.3 0.8
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Tree clustering
T1, T2, ..., Th clusters
Claim: h = W(p(X))
Y is a set of representatives if Y Å Ti · 1 for all i
Lemma: Y is ½ - flow-well-linked
Representatives are well-linked
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Representatives
Need representatives Y such that


Y ½ Xi
Y induces a large submatching of Mi
Simple greedy scheme works
pick si and ti
remove all terminals in trees of si and ti
continue
Node case
Well-linked decomposition same as for edge case
Use node-separators instead of edge separators
Clustering is not straighforward (can’t assume
degree bound)
In [CKS05] weaker bounds than for edge case
Recent work: same as for edge case. More
technically involved
Lower Bounds
Well-linked decomposition has to lose W(log1/2 n)
factor
Implicitly from integrality gap results for all-ornothing flow problem [Chuzhoy-Khanna05]
Conjecture: W(log n) factor lower bound
Flows, Cuts, and Integer Flows
NP-hard
max integer flow ·
??
Solvable
max frac flow
+ graph theory
NP-hard
·
min multicut
Flow-cut gap
thms [LR88 ...]
Weaker decomp for planar graphs
Well-linked decomp yields O(log n) approx for
planar graph EDP (congestion 2)
Recent result for planar EDP: O(1) approx with
congestion 4 [CKS 05]
Weaker decomp based on planar graph properties.
Q: well-linked in planar loses W(log n) ?
Open problems
Improve upper/lower bounds on well-linked
decomposition. Q(log n)?
Approx algorithms for EDP/NDP in general graphs
with congestion O(1)
essentially reduced to a graph theory problem
Directed graphs?
Thank You!
Trees to Graphs using Racke
Hierarchical graph decomposition [Racke03]
Given graph G, exists capacitated tree T(G) s.t
T(G) approximates G w.r.t sparse cuts
Approximation factor – O(b(G) log n log log n)
[Harrelson-Hildrum-Rao04]
Apply algo. on T(G) to get decomposition for G
Loss: polylog(n)