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 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?

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.

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)

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.

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

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

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).

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).

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.

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).

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.

The corresponding cycle in G whose weight is (1/2+1/3)/52 = 1/30. 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

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.

Example Two 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 Vi Vj

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.

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.

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.