### Every Permutation CSP of arity 3 is Approximation Resistant

Moses Charikar, Venkatesan Guruswami, Rajsekar Manokaran
2009 2009 24th Annual IEEE Conference on Computational Complexity
A permutation constraint satisfaction problem (permCSP) of arity k is specified by a subset Λ ⊆ S k of permutations on {1, 2, . . . , 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 σ of the variables that maximizes the number of constraint tuples whose ordering (under σ) follows a permutation in Λ. This is just the natural extension of constraint satisfaction
more » ... problems over finite domains (such as Boolean CSPs) to the world of ordering problems. The simplest permCSP corresponds to the case when Λ 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  . 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 Λ ⊆ Π ⊆ S3, given an instance of permCSP(Λ) that is almost-satisfiable, it is hard to find an ordering that satisfies more than |Π| 6 + ε of the constraints even under the relaxed constraint Π (for arbitrary ε > 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.