A polynomial-time algorithm for median-closed semilinear constraints

TL;DR

This paper introduces a polynomial-time algorithm for solving the CSP over semilinear constraints preserved by medians, expanding tractability beyond convex cases to include certain non-convex relations.

Contribution

It proves that median-preserved semilinear CSPs are tractable and characterizes this class as maximally tractable among semilinear relations.

Findings

Polynomial-time algorithm for median-closed semilinear CSPs

Includes non-convex relations like finite subsets and disjunctive constraints

Maximal tractability class among semilinear relations

Abstract

A subset of Q^n is called semilinear (or piecewise linear) if it is Boolean combination of linear half-spaces. We study the computational complexity of the constraint satisfaction problem (CSP) over the rationals when all the constraints are semilinear. When the sets are convex the CSP is polynomial-time equivalent to linear programming. A semilinear relation is convex if and only if it is preserved by taking averages. Our main result is a polynomial-time algorithm for the CSP of semilinear constraints that are preserved by applying medians. We also prove that this class is maximally tractable in the sense that any larger class of semilinear relations has an NP-hard CSP. To illustrate, our class contains all relations that can be expressed by linear inequalities with at most two variables (so-called TVPI constraints), but it also contains many non-convex relations, for example constraints of the form x in S for arbitrary finite subset S of Q, or more generally disjunctive constraints of the form x < c or y < d for constants c and d.