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Packing directed cycles efficiently

Packing directed cycles efficiently. Zeev Nutov Raphael Yuster. Definitions and notations. Given a digraph G , how many arc-disjoint cycles can be packed into G ? This value is the cycle packing number ν c ( G ) of G . ν c * ( G ) = max fractional cycle packing.

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Packing directed cycles efficiently

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  1. Packing directed cycles efficiently Zeev Nutov Raphael Yuster

  2. Definitions and notations • Given a digraph G, how many arc-disjoint cycles can be packed into G? This value is thecycle 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?

  3. 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.

  4. 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)

  5. Tools used - 1 Directed version of Szemeredi’s regularity lemma:(Alon and Shapira, STOC 2003). • A bipartite digraph with vertex classesA , B is called γ-regular if|d(A,B) – d(X,Y)|<γ |d(B,A) – d(Y,X)|<γfor allX 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.

  6. 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 where c(e) = number of Ck in H' containing e. j≠i

  7. Example k=3 d(3,1)=d(2,3)= ½ V1 V2 e V3

  8. 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 ythen there is a matching of size at least(q/r)(1-β).

  9. 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).

  10. 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= 0.5μδk0. • 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. • 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.

  11. 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).

  12. 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 thatν*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 andν*c)G ) ≥ w(ψ') = w(ψ*) m2/n2 ≥ (α-δ)m2.

  13. ExampleThree 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

  14. The proof cont. Use ψ' to define a random coloring of the arcs ofG*. The “colors” are the cycles of G.Let e  E(Vi,Vj) be an arc ofG*. For each cycle C inG 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.

  15. ExampleTwo cycles containing (i,j), d(i,j)=1/5 2/25 i j 3/25 In E(Vi,Vj): Prob(- - -) = 2/5 Prob(___ ) = 3/5 Vj Vi

  16. 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 random-like 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.

  17. 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)> 0all cycles C in G with ψ'(C) > m1-k0satisfy the statements of the last two lemmas. We therefore fix such a coloring.

  18. The proof cont. • Let Cbe 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 disjointCk 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 thecontribution of copies with ψ'(C) ≤ m1-k0 tow(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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