Integer programs with bounded subdeterminants and two nonzeros per row

TL;DR

This paper presents a strongly polynomial-time algorithm for certain integer linear programs with bounded subdeterminants and limited nonzeros per row, extending solutions to weighted stable set problems on specific graph classes.

Contribution

It introduces the first polynomial-time algorithm for weighted stable set problems on graphs with a fixed number of disjoint odd cycles, and applies this to solve related integer programs efficiently.

Findings

Polynomial-time algorithm for weighted stable set on graphs with bounded odd cycles

Extension of integer program solutions to matrices with two nonzeros per column

Reduction to b-matching for solving specific integer programs

Abstract

We give a strongly polynomial-time algorithm for integer linear programs defined by integer coefficient matrices whose subdeterminants are bounded by a constant and that contain at most two nonzero entries in each row. The core of our approach is the first polynomial-time algorithm for the weighted stable set problem on graphs that do not contain more than $k$ vertex-disjoint odd cycles, where $k$ is any constant. Previously, polynomial-time algorithms were only known for $k=0$ (bipartite graphs) and for $k=1$. We observe that integer linear programs defined by coefficient matrices with bounded subdeterminants and two nonzeros per column can be also solved in strongly polynomial-time, using a reduction to $b$-matching.