Truncated Markov bases and Gr\"obner bases for Integer Programming

TL;DR

This paper introduces a faster algorithm for computing truncated Markov bases and applies it to solve linear integer feasibility problems, including equality knapsack problems, with promising initial results.

Contribution

It presents a novel, more efficient algorithm for truncated Markov bases and a specialized Groebner basis approach for certain integer linear programs.

Findings

The new algorithm outperforms existing methods in speed.

The approach effectively solves equality knapsack problems.

Initial results show improved performance over previous Groebner basis methods.

Abstract

We present a new algorithm for computing a truncated Markov basis of a lattice. In general, this new algorithm is faster than existing methods. We then extend this new algorithm so that it solves the linear integer feasibility problem with promising results for equality knapsack problems. We also present a novel Groebner basis approach to solve a particular integer linear program as opposed to previous Groebner basis methods that effectively solved many different integer linear programs simultaneously. Initial results indicate that this optimisation algorithm performs better than previous Groebner basis methods.