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Approximating Maximum Subgraphs Without Short Cycles

Approximating Maximum Subgraphs Without Short Cycles. Guy Kortsarz Join work with Michael Langberg and Zeev Nutov. Max-g-Girth. Girth: A graph G is said to have girth g if its shortest cycle is of length g .

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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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