Linear Equations with Min and Max Operators: Computational Complexity

TL;DR

This paper systematically studies the computational complexity of a class of optimization problems involving linear equations with min and max operators under various restrictive conditions, revealing NP-completeness and complexity class results.

Contribution

It provides a comprehensive complexity analysis of these problems across different conditions, filling a gap in the literature.

Findings

Problems with conditions C2 and C4 are NP-complete.

Problems with conditions C3 and C4 are NP-complete.

With only condition C1, the problem is in UP intersect coUP.

Abstract

We consider a class of optimization problems defined by a system of linear equations with min and max operators. This class of optimization problems has been studied under restrictive conditions, such as, (C1) the halting or stability condition; (C2) the non-negative coefficients condition; (C3) the sum up to 1 condition; and (C4) the only min or only max oerator condition. Several seminal results in the literature focus on special cases. For example, turn-based stochastic games correspond to conditions C2 and C3; and Markov decision process to conditions C2, C3, and C4. However, the systematic computational complexity study of all the cases has not been explored, which we address in this work. Some highlights of our results are: with conditions C2 and C4, and with conditions C3 and C4, the problem is NP-complete, whereas with condition C1 only, the problem is in UP intersects coUP. Finally, we establish the computational complexity of the decision problem of checking the respective conditions.