On linear and semidefinite programming relaxations for hypergraph matching

The hypergraph matching problem is to find a largest collection of disjoint hyperedges in a hypergraph. This is a well-studied problem in combinatorial optimization and graph theory with various applications. The best known approximation algorithms for this problem are all local search algorithms. In this paper we analyze different linear and semidefinite programming relaxations for the hypergraph matching problem, and study their connections to the local search method. Our main results are the following: ? We consider the standard linear programming relaxation of the problem. We provide an algorithmic proof of a result of F?uredi, Kahn and Seymour, showing that the integrality gap is exactly k - 1 + 1/k for k-uniform hypergraphs, and is exactly k - 1 for k-partite hypergraphs. This yields an improved approximation algorithm for the weighted 3-dimensional matching problem. Our algorithm combines the use of the iterative rounding method and the fractional local ratio method, showing a ...

• Type: Conference
• Year: 2010
• Where: SODA
• Authors: Yuk Hei Chan, Lap Chi Lau
• Keywords: Discrete Algorithms, Hypergraph Matching Problem, Semidefinite Programming Relaxation, SODA 2010, Standard Lp Relaxation