# Approximating maximum subgraphs without short cycles

Guy Kortsarz, Michael Langberg, Zeev Nutov

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

## Abstract

We study approximation algorithms, integrality gaps, and hardness of approximation, of two problems related to cycles of "small" length k in a given graph. The instance for these problems is a graph G = (V,E) and an integer k. The k -Cycle Transversal problem is to find a minimum edge subset of E that intersects every k-cycle. The k -Cycle-Free Subgraph problem is to find a maximum edge subset of E without k-cycles. The 3-Cycle Transversal problem (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 Transversal admits a (2 - ε)-approximation algorithm, then so does the Vertex-Cover problem, and thus improving the ratio 2 is unlikely. We also show that k -Cycle Transversal admits a (k - 1)-approximation algorithm, which extends the result of Krivelevich from k = 3 to any k. Based on this, for odd k we give an algorithm for k -Cycle-Free Subgraph with ratio k-1/2k-3 = 1/2 + 1/4k-6; this improves over the trivial ratio of 1/2. Our main result however is for the k -Cycle-Free Subgraph problem with even values of k. For any k = 2r, we give an Ω(n - 1/r + 1/r(2r-1)-ε)-approximation scheme with running time ε-Ω(1/ε) poly(n). This improves over the ratio Ω(n -1/r ) that can be deduced from extremal graph theory. In particular, for k = 4 the improvement is from Ω(n -1/2) to Ω(1/n - 1/3 - ε ). Similar results are shown for the problem of covering cycles of length ≤ k or finding a maximum subgraph without cycles of length ≤ k.

Original language English Approximation, Randomization and Combinatorial Optimization Algorithms and Techniques - 11th International Workshop, APPROX 2008 and 12th International Workshop, RANDOM 2008, Proceedings 118-131 14 https://doi.org/10.1007/978-3-540-85363-3_10 Published - 2008 11th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2008 and 12th International Workshop on Randomization and Computation, RANDOM 2008 - Boston, MA, United StatesDuration: 25 Aug 2008 → 27 Aug 2008

### Publication series

Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) 5171 LNCS 0302-9743 1611-3349

### Conference

Conference 11th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2008 and 12th International Workshop on Randomization and Computation, RANDOM 2008 United States Boston, MA 25/08/08 → 27/08/08

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