Packing directed cycles
efficiently
Zeev Nutov
Raphael Yuster
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Definitions and notations
• Given a digraph G, how many arcdisjoint cycles can be packed into G?
This value is the
cycle packing number νc(G) of G.
• νc*(G) = max fractional cycle packing.
• Clearly νc*(G) ≤ νc(G).
• Computing νc(G) is NP-Hard.
Computing νc*(G) is in P (using LP).
• How far apart can they be?
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Main result and its algorithmic
consequences
Theorem: νc*(G) - νc(G) = o(n2).
Furthermore, a set of νc(G) - o(n2)
arc-disjoint cycles can be found in
randomized polynomial time.
Corollary: νc(G) can be approximated to
within an o(n2) additive term in
polynomial time.
This implies a FPTAS for computing
νc(G) for almost all digraphs, since
νc(G) = θ(n2) for almost all digraphs.
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A more general result
• Let F be any fixed (finite or infinite)
family of oriented graphs.
ν(F,G) = max F–packing value in G.
ν*(F,G) = max fractional F–packing.
Theorem: ν*(F,G) - ν(F,G) =o(n2).
• Our result for cycles follows by letting
F be the family of all cycles and from
the fact that all 2-cycles appear in any
max cycle packing.
• The special case where F is a single
undirected graph has been proved by
Haxell and Rödl (Combinatorica 2001)
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Tools used - 1
Directed version of Szemeredi’s regularity lemma:
(Alon and Shapira, STOC 2003).
– A bipartite digraph with vertex classes
A , B is called γ-regular if
|d(A,B) – d(X,Y)|<γ |d(B,A) – d(Y,X)|<γ
for all X A, |X| > γ|A|, Y B, |Y| > γ|B|,
where d(.,.) is the arc density of the pair.
– A γ-regular partition of V is an equitable
partition such that all (but a γ-fraction) of the
part pairs are γ-regular.
– For every γ>0, there is an integer M(γ)>0
such that every digraph G of order n > M
has a γ-regular partition of its vertex set
into m parts, for some 1/γ < m < M.
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Tools used - 2
A “random-like behavior lemma”:
For reals δ , ζ and positive integer k there exist
γ = γ(δ, ζ, k) and T=T(δ, ζ, k) such that:
Any k-partite oriented graph H with parts
V1,…,Vk with |Vi|=t >T that satisfies:
- each pair (Vi,Vi+1) is γ-regular;
- d(Vi,Vi+1) > δ,
has a spanning subgraph H' with at least
(1-ζ)|E(H)| arcs such that for e E(Vi,Vi+1)
|c(e)/tk-2 - ∏d(j,j+1)| < ζ
j≠i
where c(e) = number of Ck in H' containing e.
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k=3
Example
d(3,1)=d(2,3)= ½
V1
V2
e
V3
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Tools used – 3
Frankl-Rödl hypergraph matching theorem:
For an integer r > 1 and a real β > 0 there
exists a real μ > 0 so that if an r-uniform
hypergraph on q vertices has the following
properties for some d:
(i) (1- μ)d < deg(x) < (1+ μ)d for all x
(ii) deg(x,y) < μd for all distinct x and y
then there is a matching of size at least
(q/r)(1-β).
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Tools used – 4
Theorem: A maximum fractional dicycle
packing of G yielding νc*(G) can be
computed in polynomial time.
Remark: Computing νc*(G) is in P via solving
the dual LP. But finding an appropriate
weight function w on the cycle set of G is not
straightforward (there is always an optimal
fractional packing in which only O(n2) cycles
receive nonzero weight).
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The proof
• Let ε > 0. We shall prove:
There exists N=N(ε) such that for all n > N,
if G is an n-vertex oriented graph then
ν*c(G) - νc(G) < εn2.
• A consistent “horrible” parameter selection:
– k0=20/ε (“long” cycles are ignored)
– δ=β=ε/4.
– μ=μ(β,k0) of Frankl-Rödl.
– ζ= 0.5μδk .
– γ=γ(δ, ζ ,k0) T=T(δ, ζ ,k0) as in the
“random-like behavior lemma”.
– M=M(γε/25k0) as in regularity lemma.
– N = suff. large w.r.t. these parameters.
0
• Fix an n-vertex oriented graph G with n > N.
Let ψ be a fractional dicycle packing with
w(ψ)= ν*c(G) = αn2 > εn2.
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The proof cont.
• Apply directed regularity lemma to G and
obtain a γ' -regular partition with m' parts,
where γ' =γε/(25k0) and 1/γ' < m' < M(γ').
• Refine the partition by randomly partitioning
each part into 25k0/ε parts.
The refined partition is now γ-regular.
What we gain: with positive probability the
contribution of bad cycles (cycles with two
vertices in the same vertex class) to w(ψ) is
less than εn2/20. We may therefore assume
that there are no bad cycles.
• Let V1,…,Vm be the vertex classes of the
refined partition, m = m' (25k0/ε).
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The proof cont.
• Let G* be the spanning subgraph of G
consisting of the arcs connecting part pairs
that are γ-regular and with density > δ.
• Let ψ* be the restriction of ψ to G* (namely,
“surviving” cycles). It is easy to show that
ν*c(G*) ≥ w(ψ*) > w(ψ)- δn2 = (α-δ)n2.
• Let G be the m-vertex super-digraph obtained
from G* by contracting each part.
Define a fractional packing ψ' of G by
“gluing” parallel cycles and scaling by m2 / n2.
• Observation: ψ' is proper and
ν*c(G ) ≥ w(ψ') = w(ψ*) m2/n2 ≥ (α-δ)m2.
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Example
Three parts, n/m=5, two “parallel” cycles in G*
having weights 1/2 and 1/3.
1/2
1/3
The corresponding cycle in G whose weight
is (1/2+1/3)/52 = 1/30.
1/30
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The proof cont.
Use ψ' to define a random coloring of the
arcs of G*. The “colors” are the cycles of G.
Let e  E(Vi,Vj) be an arc of G*. For each
cycle C in G that contains the arc (i,j), e is
colored “C” with probability ψ'(C)/d(i,j).
– The choices made for distinct arcs of G*
are independent.
– The random coloring is probabilistically
sound as ψ' is a proper fractional packing.
Thus S{ψ'(C): (i,j)C} ≤ d(i,j) ≤ 1.
– Some arcs might stay uncolored.
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Example
Two cycles containing (i,j), d(i,j)=1/5
2/25
i
j
3/25
In E(Vi,Vj):
Vi
Prob(- - -) = 2/5
Prob(___ ) = 3/5
Vj
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The proof cont.
• Let C ={1,…,k} in G with ψ'(C) > m1-k.
Let GC = G*[V1,…,Vk].
– GC satisfies the conditions of the randomlike behavior lemma.
– Let G'C be the spanning subgraph of GC
with properties guaranteed by the lemma.
– Let JC denote the random spanning
subgraph of GC consisting only of the arcs
whose “color” is C.
– For an arc e  E(JC), let cC(e) be the
number of Ck copies in JC containing e.
Lemma: Let eE(JC). With probability > 1-m3/n
| cC(e)/tk-2 - ψ'(C)k-1 | < μ ψ'(C)k-1.
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The proof cont.
• We also need a lower bound for the number
of arcs of JC :
With probability at least 1-1/n,
|E(JC)| > k(1-2ζ) ψ'(C) n2/m2.
• Since there are at most O(mk0) cycles in G we
have that with probability at least
1-O(mk0/n) – O(mk0+3/n) > 0
all cycles C in G with ψ'(C) > m1-k0 satisfy
the statements of the last two lemmas.
We therefore fix such a coloring.
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The proof cont.
• Let C be a k-cycle in G with ψ'(C) > m1-k0.
We construct a k-uniform hypergraph HC:
– The vertices of HC are the arcs of JC.
– The edges of HC are the arc sets of the
copies of Ck in JC .
• Our hypergraph satisfies the FR theorem
with d=tk-2 ψ'(C)k-1.
• By FR: (q/k)(1-β) arc disjoint Ck in JC.
As q > k(1-2ζ) ψ'(C) n2 / m2 we have
(1-β) (1-2ζ) ψ'(C) n2 /m2 ≥ (1-2β)ψ'(C)n2 /m2.
• Recall that w(ψ') ≥ m2(α-δ). Since the
contribution of copies with ψ'(C) ≤ m1-k0 to
w(ψ') is < m, summing the last inequality
over all cycles C with ψ'(C) > m1-k0 we have
at least (α-ε)n2 arc disjoint cycles in G.
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