Approximating Bounded Occurrence Ordering CSPs

Abstract

A theorem of Håstad shows that for every constraint satisfaction problem (CSP) over a fixed size domain, instances where each variable appears in at most O(1) constraints admit a non-trivial approximation algorithm, in the sense that one can beat (by an additive constant) the approximation ratio achieved by the naive algorithm that simply picks a random assignment. We consider the analogous question for ordering CSPs, where the goal is to find a linear ordering of the variables to maximize the number of satisfied constraints, each of which stipulates some restriction on the local order of the involved variables. It was shown recently that without the bounded occurrence restriction, for every ordering CSP it is Unique Games-hard to beat the naive random ordering algorithm.

In this work, we prove that the CSP with monotone ordering constraints $x_{i_1} < x_{i_2} < \cdots < x_{i_k}$ of arbitrary arity k can be approximated beyond the random ordering threshold 1/k! on bounded occurrence instances. We prove a similar result for all ordering CSPs, with arbitrary payoff functions, whose constraints have arity at most 3. Our method is based on working with a carefully defined Boolean CSP that serves as a proxy for the ordering CSP. One of the main technical ingredients is to establish that certain Fourier coefficients of this proxy constraint have substantial mass. These are then used to guarantee a good ordering via an algorithm that finds a good Boolean assignment to the variables of a low-degree bounded occurrence multilinear polynomial. Our algorithm for the latter task is similar to Håstad’s earlier method but is based on a greedy choice that achieves a better performance guarantee.

This research was supported in part by NSF CCF 1115525 and US-Israel BSF grant 2008293. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. A full version of this paper can be found hat http://eccc.hpi-web.de/report/2012/074/