Approximating Maximum Subgraphs without Short Cycles

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-CycleFree 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-)-approximation algorithm, then so does the Vertex-Cover problem, and thus improving the ratio 2 is unlikely. We also show that k-Cycle Transversal ...

• Type: Conference
• Year: 2008
• Where: APPROX
• Authors: Guy Kortsarz, Michael Langberg, Zeev Nutov
• Keywords: Algorithms, APPROX 2008, Integrality Gap, Subgraph Problem, Transversal Problem