Approximating Maximum Subgraphs without Short Cycles

• Authors: Guy Kortsarz;Michael Langberg;Zeev Nutov
• Venue: APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
• Year: 2008

We study approximation algorithms, integrality gaps, and hardness of approximation, of two problems related to cycles of "small" length kin a given graph. The instance for these problems is a graph G= (V,E) and an integer k. The k-Cycle Transversalproblem is to find a minimum edge subset of Ethat intersects every k-cycle. The k-Cycle-Free Subgraphproblem is to find a maximum edge subset of Ewithout k-cycles.The 3-Cycle Transversalproblem (covering all triangles) was studied by Krivelevich [Discrete Mathematics, 1995], where an LP-based 2-approximation algorithm was presented. The integrality gap of the underlying LP was posed as an open problem in the work of Krivelevich. We resolve this problem by showing a sequence of graphs with integrality gap approaching 2. In addition, we show that if 3-Cycle Transversaladmits a (2 茂戮驴 茂戮驴)-approximation algorithm, then so does the Vertex-Coverproblem, and thus improving the ratio 2 is unlikely. We also show that k-Cycle Transversaladmits a (k茂戮驴 1)-approximation algorithm, which extends the result of Krivelevich from k= 3 to any k. Based on this, for odd kwe give an algorithm for k-Cycle-Free Subgraphwith ratio $\frac{k-1}{2k-3}=\frac{1}{2}+\frac{1}{4k-6}$; this improves over the trivial ratio of 1/2.Our main result however is for the k-Cycle-Free Subgraphproblem with even values of k. For any k= 2r, we give an $\Omega\left(n^{-\frac{1}{r}+\frac{1}{r(2r-1)}-\varepsilon}\right)$-approximation scheme with running time 茂戮驴茂戮驴 茂戮驴(1/茂戮驴)poly(n). This improves over the ratio 茂戮驴(n茂戮驴 1/r) that can be deduced from extremal graph theory. In particular, for k= 4 the improvement is from 茂戮驴(n茂戮驴 1/2) to 茂戮驴(1/n茂戮驴 1/3 茂戮驴 茂戮驴).Similar results are shown for the problem of covering cycles of length ≤ kor finding a maximum subgraph without cycles of length ≤ k.