Polyhedral characteristics of balanced and unbalanced bipartite subgraph problems

TL;DR

This paper investigates the polyhedral structures of three bipartite subgraph problems, revealing their NP-hardness and providing bounds on their associated graph properties that influence algorithmic complexity.

Contribution

It characterizes the polytopes and cone decompositions of balanced and unbalanced biclique problems, offering new insights into their computational complexity.

Findings

NP-hardness of all three problems established

Superpolynomial lower bounds on clique numbers derived

Adjacency criteria for the polytope of the balanced biclique problem described

Abstract

We study the polyhedral properties of three problems of constructing an optimal complete bipartite subgraph (a biclique) in a bipartite graph. In the first problem we consider a balanced biclique with the same number of vertices in both parts and arbitrary edge weights. In the other two problems we are dealing with unbalanced subgraphs of maximum and minimum weight with nonnegative edges. All three problems are established to be NP-hard. We study the polytopes and the cone decompositions of these problems and their 1-skeletons. We describe the adjacency criterion in 1-skeleton of the polytope of the balanced complete bipartite subgraph problem. The clique number of 1-skeleton is estimated from below by a superpolynomial function. For both unbalanced biclique problems we establish the superpolynomial lower bounds on the clique numbers of the graphs of nonnegative cone decompositions. These values characterize the time complexity in a broad class of algorithms based on linear comparisons.