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? Max number of arc-disjoint cycles in G = cycle packing number ν c (G) of G. ν c *(G) = max fractional cycle packing. Clearly ν c *(G) ≤ ν c (G). Computing ν c (G) is NP-Hard. Computing ν c *(G) is in P (using LP). How far apart can they be?

3 Main result and its algorithmic consequences Theorem: ν c *(G) - ν c (G) = o(n 2 ). Furthermore, a set of ν c (G) - o(n 2 ) arc-disjoint cycles can be found in randomized polynomial time. Corollary: ν c (G) can be approximated to within an o(n 2 ) additive term in randomized polynomial time. This implies a FPTAS for computing ν c (G) for almost all digraphs, since ν c (G) = θ(n 2 ) 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(n 2 ). 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 parts A,B is γ-regular if: |d(A,B) – d(X,Y)| γ|A|, Y  B, |Y| > γ|B|, where d(.,.) is the arc density of the pair. –A γ-regular partition of V is an (almost) 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 V 1,…,V k with |V i |=t >T that satisfies: - each pair (V i,V i+1 ) is γ-regular, and - d(V i,V i+1 ) > δ, has a spanning subgraph H' with at least (1-ζ)|E(H)| arcs such that for e  E(V i,V i+1 ) |c(e)/t k-2 - ∏d(j,j+1)| < ζ where c(e) = number of C k in H' containing e.

7 V1V1 V2V2 V3V3 e Example k=3 d(3,1)=d(2,3)= ½

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 a k-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/k)(1-β).

9 Tools used – 4 Theorem: A maximum fractional dicycle packing of G yielding ν c *(G) can be computed in polynomial time. Remark: Computing ν c *(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(n 2 ) 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) - ν c (G) < εn 2. A consistent “horrible” parameter selection: –k 0 =20/ε (“long” cycles are ignored) –β=ε/4, μ=μ(β,k 0 ) of Frankl-Rödl. –δ=ε/4, ζ= 0.5μδ k 0. –γ=γ(δ, ζ,k 0 ) T=T(δ, ζ,k 0 ) as in the “random-like behavior lemma”. –M=M(γε/25k 0 ) 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) = αn 2 > εn 2.

11 The proof cont. Apply the directed regularity lemma to G and obtain a γ' -regular partition with m' parts, where γ' =γε/(25k 0 ) and 1/γ' < m' < M(γ'). Refine the partition by randomly partitioning each part into 25k 0 /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 part) to w(ψ) is less than εn 2 /20. We may therefore assume that there are no such cycles. Let V 1,…,V m be the vertex classes of the refined partition, m = m' (25k 0 /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 that ν* c )G*) ≥ w(ψ*) > w(ψ)- δn 2 = (α-δ)n 2. Let  be the m-vertex super-digraph obtained from G* by contracting each part. Define a fractional packing ψ' of  by “gluing parallel cycles” and scaling by m 2 / n 2. Observation: ψ' is a proper packing and ν* c )  ) ≥ w(ψ') = w(ψ*) m 2 /n 2 ≥ (α-δ)m 2.

13 1/2 1/3 Example Three parts, n/m=5, two “parallel” cycles in G* having weights 1/2 and 1/3. The packing in  consists of a cycle of the weight (1/2+1/3)/5 2 = 1/30. 1/30

14 The proof cont. Use ψ' to define a random coloring of the arcs of G*. The “colors” are the cycles of . Let e  E(V i,V j ) be an edge of G*. For each cycle C in  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  {ψ'(C): (i,j)  C} ≤ d(i,j). –Some arcs might stay uncolored.

15 ji 2/25 3/25 ViVi VjVj Example Two cycles containing (i,j), d(i,j)=1/5 In E(V i,V j ): Prob(- - -) = 2/5 Prob( ___ ) = 3/5

16 The proof cont. Let C ={1,…,k} in  with ψ'(C) > m 1-k. Let G C = G*[V 1,…,V k ]. –G C satisfies the conditions of the random- like behavior lemma. –Let G' C be the spanning subgraph of G C with properties guaranteed by the lemma. –Let J C denote the random spanning subgraph of G C consisting only of the arcs whose “color” is C. –For an arc e  E(J C ), let c(e) be the number of C k copies in J C containing e. Put r=k. Lemma: Let e  E(J C ). With probability > 1-m 3 /n | c(e)/t k-2 - ψ'(C) r-1 | < μ ψ'(C) r-1.

17 The proof cont. A lower bound for the number of arcs of J C : With probability at least 1-1/n: |E(J C )| > r(1-2ζ) ψ ' (C) n 2 /m 2. Since there are at most O(m k 0 ) cycles in  we have that with probability at least 1-O(m k 0 /n) – O(m k 0 +3 /n) > 0 all cycles C in  with ψ'(C) > m 1-k 0 satisfy the statements of the last two lemmas. We therefore fix such a coloring.

18 The proof cont. Let C k in  with ψ'(H) > m 1-k 0. Construct an r-uniform hypergraph H C : –The vertices of H C are the arcs of J C. –The edges of H C are the arc sets of the copies of C k in J C. Our hypergraph satisfies the FR theorem with d=t k-2 ψ'(C) r-1. By FR: (q/r)(1-β) arc disjoint C k in J C. As q > r(1-2ζ) ψ'(C) n 2 / m 2 we have (1-β) (1-2ζ) ψ'(C) n 2 /m 2 ≥ (1-2β)ψ'(C)n 2 /m 2. Recall that w(ψ') ≥ m 2 (α-δ). Since the contribution of copies with ψ'(C) ≤ m 1-k 0 to w(ψ') is m 1-k 0 we have at least (α-ε)n 2 arc disjoint cycles in G.