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Approximating Maximum Subgraphs Without Short Cycles

This paper presents the Max-g-Girth problem—finding subgraphs of a given girth with maximum edges. We explore the challenges when g=4 (triangle-free) and the complexity of approximation, demonstrating the NP-Hard nature of the problem. Notably, our results improve upon trivial ratios for various g values and propose a framework for optimizing Max-5-Girth on bipartite graphs. The study also discusses cycle covering problems and establishes connections to integrality gaps. We highlight remarkable open problems and theories related to approximation ratios, contributing to the ongoing exploration of graph theory.

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Approximating Maximum Subgraphs Without Short Cycles

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  1. Approximating Maximum Subgraphs WithoutShort Cycles Guy Kortsarz Join work with Michael Langberg and Zeev Nutov

  2. Max-g-Girth Girth: A graph G is said to have girth g if its shortest cycle is of length g. Max-g-Girth: Given G, find a subgraph of G of girth at least g with the maximum number of edges. g=4

  3. Max-g-Girth: context • Max-g-Girth: • Used in study of “Genome Sequencing” [Pevzner Tang Tesler]. • Mentioned in [ErdosGallaiTuza] for g=4 (triangle free). • Complementary problem of “covering” all small cycles (size ≤ g) with minimum number of edges was studied in past. • [Krivelevich] addressed g=4 (covering triangles). • Approximation ratio of 2 was achieved (ratio of 3 is easy). • Problem is NP-Hard (even for g=4).

  4. Max-g-Girth on cliques The Max-g-Girth problem on cliques = densest graph on n vertices with girth g. Has been extensively studied [Erd¨os, BondySimonovits, …]]. Known that Max-g-Girth has size between (n1+4/(3g-12)) and O(n1+2/(g-2)). There is a polynomial gap! Long standing open problem. Implies that approximation ratio O(n-) will solve open problem.

  5. First steps • Positive: • Trivial by previous bounds approximation ratio of ~ n-2/(g-2). • For g=5,6  n-1/2. • If g>4 part of input: ratio n-1/2. • If g=4 (maximum triangle free graph): return random cut and obtain ½|EG| edgesratio ½. • g = 4: constant ratio, g ≥ 5 polynomial ratio!

  6. Our results • Max-g-Girth: positive and negative. • Positive: • Improve on trivial n-1/2 for general g to n-1/3. • For g=4 (triangle free) improve from ½ to 2/3. • For instances withn2edges: ratio ~ n-2/3g. • Negative: • Max-g-Girth is APX hard (any g). Large gap!

  7. Our results • Covering triangles by edges. • [Krivelevich] presented LP based 2 approx. ratio. • Posed open problem of tightness of integrality gap. • We solve open problem: present family of graphs in which the gap is 2-. • Moreover: 2- approximation implies 2- for Vertex Cover (<1/2).

  8. Positive • Theorem: Max-g-Girth admits ratio ~ n-1/3. • Outline of proof: • Consider optimal subgraphH. • Remove all odd cycles in G by randomly partitioning G and removing edges on each side. • ½ the edges of optimal H remain  Opt. value “did not” change. • Now G is bipartite, need to remove even cycles of size < g. • If g=5: only need to remove cycles of length 4. • If g=6: only need to remove cycles of length 4. • If g>6: asany graph of girth g=2r+1 or 2r+2 contains at most ~ n1+1/r edges, trivial algorithm gives ratio n-1/3. • Goal: Approximate Max-5-Girth within ratio ~ n-1/3.

  9. Max-5-Girth Step I: • General procedure that may be useful elsewhere. • Let G=(A,B;E) – want G’ almost regular on A and B & Opt(G’)~Opt(G). • Starting point: easy to make A regular (bucketing). • Now we can make B regular, howeverA becomes irregular. • Iterate … • Can show: if we do not converge after constant # steps then it must be the case that Opt(G) is small (in each iteration degree decreases). • Goal: App. Max-5-Girth on bipartite graphs within ratio ~ n-1/3. • Namely: given bipartite G find max. HG without 4-cycles. • Algorithm has 2 steps: • Step I: Find G’G that is almost regular (in both parts) such that Opt(G’)~Opt(G). • Step II: Find HG’ for which |EH| ≥ Opt(G’)n-1/3.

  10. Max-5-girth Step II: • Now G’ is regular. • Enables us to tightly analyze the maximum amount of 4 cycles in G’. Regularity connects # of edges |EG’| with number of 4-cycles. • Remove edges randomly as to break 4-cyles (“alteration method”). • Using comb. upper bound on Opt[NaorVerstraete] yields n-1/3 ratio. 10 • Goal: App. Max-5-girth on bipartite graphs within ratio ~ n-1/3. • Namely: given bipartite G find max. HG without 4-cycles. • Algorithm has 2 steps: • Step I: Find G’G that is almost regular (in both parts) such that Opt(G’)~Opt(G). • Step II: Find HG’ for which |EH|≥ Opt(G’)n-1/3.

  11. Covering k cycles Our algorithm actually gives an approximation for the problem of finding a maximum edge subgraph of G without cycles of length exactly k. Trivial algorithm (return spanning tree) gives ratio of n-2/k Our algorithm gives ~ n-2/k (1+1/(k-1)) Significant for small values of k.

  12. Some interesting open prob. • LP for g=4 (maximum triangle free graph): Max:ex(e) st:For every triangle C, eCx(e)2 • Max-Cut: integrality gap = 2. • Complete graph: IG = 4/3 (x(e)=2/3). • Conjecture: NP-Hard to obtain 2/3+ approx.

  13. Some interesting open prob. Max-5-girth: • Large gap between upper and lower bounds. • We suspect that for some  a ratio of n- isNP-Hard. • Obvious open problem: give strong lower bound for g=5.

  14. Some interesting open prob. Thanks! Set Cover in which each element appears in k sets. • Upper bound: k. • Lower bound: k-1-[Dinur et al.] • If sets are “k cycles” in given graph G we show a ratio of k-1 (for odd k). • Open problem: is k-1 possible for general set cover (large k).

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