## 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 |
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Title of host publication | Approximation, Randomization and Combinatorial Optimization |

Subtitle of host publication | Algorithms and Techniques - 11th International Workshop, APPROX 2008 and 12th International Workshop, RANDOM 2008, Proceedings |

Pages | 118-131 |

Number of pages | 14 |

DOIs | |

State | Published - 2008 |

Event | 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 States Duration: 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) |
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Volume | 5171 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 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 |
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Country/Territory | United States |

City | Boston, MA |

Period | 25/08/08 → 27/08/08 |