A polynomial Time Algorithm to Solve The Max-atom Problem

TL;DR

This paper introduces a polynomial-time algorithm for the Max-atom problem, a class of inequalities involving maxima, showing that several related problems are also in P, thus advancing computational complexity understanding.

Contribution

The paper presents the first polynomial-time algorithm for the Max-atom problem, expanding the class of problems known to be solvable efficiently.

Findings

Max-atom problem is in P with an $O(n^{6} m^{2} + n^{4} m^{3} + n^{2} m^{4})$ algorithm.

Several problems previously in NP ∩ co-NP are now shown to be in P.

The algorithm enables efficient solutions for related problems in tropical geometry, game theory, and scheduling.

Abstract

In this paper we consider $m$ ($m \geq 1$)conjunctions of Max-atoms that is atoms of the form $\max(z,y) + r \geq x$, where the offset $r$ is a real constant and $x,y,z$ are variables. We show that the Max-atom problem (MAP) belongs to $\textsf{P}$. Indeed, we provide an algorithm which solves the MAP in $O(n^{6} m^{2} + n^{4} m^{3} + n^{2} m^{4})$ operations, where $n$ is the number of variables which compose the max-atoms. As a by-product other problems also known to be in $\textsf{NP} \cap \textsf{co-NP}$ are in $\textsf{P}$. P1: the problem to know if a tropical cone is trivial or not. P2: problem of tropical rank of a tropical matrix. P3: parity game problem. P4: scheduling problem with AND/OR precedence constraints. P5: problem on hypergraph (shortest path). P6: problem in model checking and $\mu$-calculus.