Satisfiability of Ordering CSPs Above Average

TL;DR

This paper proves that the problem of determining satisfiability above average for ordering CSPs of any arity is fixed-parameter tractable, extending previous results and introducing a new inequality for functions on product spaces.

Contribution

It establishes fixed-parameter tractability for all arities of ordering CSPs above average and generalizes to CSPs with linear inequality predicates, along with a novel Bonami-type inequality.

Findings

Proved fixed-parameter tractability for all arities of ordering CSPs.

Generalized results to CSPs with predicates defined by linear inequalities.

Developed a new Bonami-type inequality applicable to arbitrary product spaces.

Abstract

We study the satisfiability of ordering constraint satisfaction problems (CSPs) above average. We prove the conjecture of Gutin, van Iersel, Mnich, and Yeo that the satisfiability above average of ordering CSPs of arity $k$ is fixed-parameter tractable for every $k$. Previously, this was only known for $k=2$ and $k=3$. We also generalize this result to more general classes of CSPs, including CSPs with predicates defined by linear inequalities. To obtain our results, we prove a new Bonami-type inequality for the Efron-Stein decomposition. The inequality applies to functions defined on arbitrary product probability spaces. In contrast to other variants of the Bonami Inequality, it does not depend on the mass of the smallest atom in the probability space. We believe that this inequality is of independent interest.