The Maximum Induced Bipartite Subgraph Problem with Edge Weights
Denis Cornaz, Ali Ridha Mahjoub · SIAM Journal on Discrete Mathematics · 2007
Given a graph $G=(V,E)$ with nonnegative weights on the edges, the maximum induced bipartite subgraph problem (MIBSP) is to find a maximum weight bipartite subgraph $(W,E[W])$ of G. Here $E[W]$ is the edge set induced by W. An edge subset $F\subseteq E$ is called independent if there is an induced bipartite subgraph of G whose edge set contains F. Otherwise, it is called dependent. In this paper we characterize the minimal dependent sets, that is, the dependent sets that are not contained in any other dependent set. Using this, we give an integer linear programming formulation for MIBSP in the natural variable space, based on an associated class of valid inequalities called dependent set inequalities. Moreover, we show that the minimum dependent set problem with nonnegative weights can be reduced to the minimum circuit problem in a directed graph, and can then be solved in polynomial time. This yields a polynomial-time separation algorithm for the dependent set inequalities as well as a polynomial-time cutting plane algorithm for solving the linear relaxation of the problem. We also discuss some polyhedral consequences.