Finding total unimodularity in optimization problems solved by linear programs

TL;DR

This paper investigates conditions under which linear programming relaxations of combinatorial optimization problems yield integer solutions, focusing on total unimodularity, and proposes methods to efficiently find optimal solutions.

Contribution

It introduces a decomposition approach for certain problems into subproblems, utilizing total unimodularity to develop faster combinatorial algorithms.

Findings

Total unimodularity enables efficient solutions for specific subproblems.

Decomposition into slot selection and assignment simplifies complex problems.

Proposed algorithms improve worst-case running time.

Abstract

A popular approach in combinatorial optimization is to model problems as integer linear programs. Ideally, the relaxed linear program would have only integer solutions, which happens for instance when the constraint matrix is totally unimodular. Still, sometimes it is possible to build an integer solution with the same cost from the fractional solution. Examples are two scheduling problems and the single disk prefetching/caching problem. We show that problems such as the three previously mentioned can be separated into two subproblems: (1) finding an optimal feasible set of slots, and (2) assigning the jobs or pages to the slots. It is straigthforward to show that the latter can be solved greedily. We are able to solve the former with a totally unimodular linear program, from which we obtain simple combinatorial algorithms with improved worst case running time.