Every Permutation CSP of arity 3 is Approximation Resistant

Abstract

A permutation constraint satisfaction problem (permCSP) of arity k is specified by a subset Lambda of permutations on $\{1,2,\dots,k\}$. An instance of such a permCSP consists of a set of variables $V$ and a collection of constraints each of which is an ordered $k$-tuple of $V$. The objective is to find a global ordering $\sigma$ of the variables that maximizes the number of constraint tuples whose ordering (under $\sigma$) follows a permutation in $\Lambda$. This is just the natural extension of constraint satisfaction problems over finite domains (such as Boolean CSPs) to the world of ordering problems. The simplest permCSP corresponds to the case when $\Lambda$ consists of the identity permutation on two variables. This is just the Maximum Acyclic Subgraph (\mas) problem. It was recently shown that the \mas\ problem is Unique-Games hard to approximate within a factor better than the trivial $1/2$ achieved by a random ordering [GMR08]. Building on this work, in this paper we show that for *every* permCSP of arity $3$, beating the random ordering is Unique-Games hard. The result is in fact stronger: we show that for every $\Lambda \subseteq \Pi \subseteq S_3$, given an instance of permCSP$(\Lambda)$ that is almost-satisfiable, it is hard to find an ordering that satisfies more than $\frac{|\Pi|}{6} +\eps$ of the constraints even under the relaxed constraint $\Pi$ (for arbitrary $\eps ≫ 0$). A special case of our result is that the *Betweenness* problem is hard to approximate beyond a factor $1/3$. Interestingly, for *satisfiable* instances of Betweenness, a factor $1/2$ approximation algorithm is known. Thus, every permutation CSP of arity up to $3$ resists approximation beyond the trivial random ordering threshold. In contrast, for Boolean CSPs, there are both approximation resistant and non-trivially approximable CSPs of arity $3$.