# The *t*-vertex cover problem: Extending the half integrality framework with budget constraints

## Abstract

In earlier work we defined a class of integer programs with constraints that involve up to three variables each and showed how to derive superoptimal half integral solution to such problems. These solutions can be used under certain conditions to generate 2-approximations. Here we extend these results to problems involving budget constraints that do not conform to the structure of that class. Specifically, we address the *t*-vertex cover problem recently studied in the literature. In this problem the aim is to cover at least *t* edges in the graph with minimum weight collection of vertices that are adjacent to these edges.

The technique proposed employs a relaxation of the budget constraint and a search for optimal dual multiplier assigned to this constraint. The multipliers can be found substantially more efficiently than with approaches previously proposed that require the solution of a linear programming problem using the interior point or ellipsoid method. Instead of linear programming we use a combinatorial algorithm solving the minimum cut problem.

## Keywords

Budget Constraint Cover Problem Vertex Cover Linear Programming Relaxation Vertex Cover Problem## Preview

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