The generalized vertex cover problem and some variations

TL;DR

This paper explores the generalized vertex cover problem (GVC), establishing its equivalences to other optimization problems, identifying solvable cases, and proposing approximation algorithms, especially on bipartite graphs.

Contribution

It introduces the GVC as a unifying framework, analyzes its properties, and provides new polynomially solvable cases and approximation strategies for specific instances.

Findings

GVC is equivalent to unconstrained binary quadratic programming.

Identified solvable cases and approximation algorithms for GVC.

GVC on bipartite graphs is equivalent to bipartite unconstrained quadratic programming.

Abstract

In this paper we study the generalized vertex cover problem (GVC), which is a generalization of various well studied combinatorial optimization problems. GVC is shown to be equivalent to the unconstrained binary quadratic programming problem and also equivalent to some other variations of the general GVC. Some solvable cases are identified and approximation algorithms are suggested for special cases. We also study GVC on bipartite graphs and identify some polynomially solvable cases. We show that GVC on bipartite graphs is equivalent to the bipartite unconstrained 0-1 quadratic programming problem. Integer programming formulations of GVC and related problems are presented and establish half-integrality property on some variables for the corresponding linear programming relaxations. We also discuss special cases of GVC where all feasible solutions are independent sets or vertex covers. These problems are observed to be equivalent to the maximum weight independent set problem or minimum weight vertex cover problem along with some algorithmic results.