Faster Algorithms for Sparse ILP and Hypergraph Multi-Packing/Multi-Cover Problems

TL;DR

This paper introduces faster algorithms for solving sparse integer linear programming problems and various multi-packing/multi-cover problems on graphs and hypergraphs, improving efficiency on sparse instances.

Contribution

It presents new exponential algorithms for ILP and counting problems on sparse matrices, and extends these methods to a broad class of multi-packing and multi-cover problems.

Findings

Algorithms outperform existing ILP and counting methods on sparse instances.

New exponential algorithms for edge/vertex multi-packing/multi-cover problems.

Framework covers many generalizations of classical combinatorial problems.

Abstract

In our paper, we consider the following general problems: check feasibility, count the number of feasible solutions, find an optimal solution, and count the number of optimal solutions in $P \cap Z^n$, assuming that $P$ is a polyhedron, defined by systems $A x \leq b$ or $Ax = b,\, x \geq 0$ with a sparse matrix $A$. We develop algorithms for these problems that outperform state of the art ILP and counting algorithms on sparse instances with bounded elements. We use known and new methods to develop new exponential algorithms for Edge/Vertex Multi-Packing/Multi-Cover Problems on graphs and hypergraphs. This framework consists of many different problems, such as the Stable Multi-set, Vertex Multi-cover, Dominating Multi-set, Set Multi-cover, Multi-set Multi-cover, and Hypergraph Multi-matching problems, which are natural generalizations of the standard Stable Set, Vertex Cover, Dominating Set, Set Cover, and Maximal Matching problems.