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Abstract. We prove that, assuming the Unique Games conjecture (UGC), every problem in the class of ordering constraint satisfaction problems (OCSPs) where each constraint has constant arity is approximation resistant. In other words, we show that if ρ is the expected fraction of constraints satisfied by a random ordering, then obtaining a ρ approximation for any ρ > ρ is UG-hard. For the simplest OCSP, the Maximum Acyclic Subgraph (MAS) problem, this implies that obtaining a ρ-approximation for any constant ρ > 1/2 is UG-hard. Specifically, for every constant ε > 0 the following holds: given a directed graph G that has an acyclic subgraph consisting of a fraction (1 − ε) of its edges, it is UG-hard to find one with more than (1/2 + ε) of its edges. Note that it is trivial to find an acyclic subgraph with 1/2 the edges by taking either the forward or backward edges in an arbitrary ordering of the vertices of G. The MAS problem has been well studied, and beating the random ordering for MAS has been a basic open problem. An OCSP of arity k is specified by a subset Π ⊆ S k of permutations on {1, 2, . . . , k}. An instance of such an OCSP is a set V and a collection of constraints, each of which is an ordered k-tuple of V . The objective is to find a global linear ordering of V while maximizing the number of constraints ordered as in Π. A random ordering of V is expected to satisfy a ρ = |Π| k! fraction. We show that, for any fixed k, it is hard to obtain a ρ -approximation for Π-OCSP for any ρ > ρ. The result is in fact stronger: we show that for every Λ ⊆ Π ⊆ S k , and an arbitrarily small ε, it is hard to distinguish instances where a (1 − ε) fraction of the constraints can be ordered according to Λ from instances where at most a (ρ + ε) fraction can be ordered as in Π. A special case of our result is that the Betweenness problem is hard to approximate beyond a factor 1/3. The results naturally generalize to OCSPs which assign a payoff to the different permutations. Finally, our results imply (unconditionally) that a simple semidefinite relaxation for MAS does not suffice to obtain a better approximation.

In a beautiful result, Raghavendra established optimal Unique Games Conjecture (UGC)-based inapproximability for a large class of constraint satisfaction problems (CSPs). In the class of CSPs he considers, of which Maximum Cut is a prominent example, the goal is to find an assignment which maximizes a weighted fraction of constraints satisfied. He gave a generic semi-definite program (SDP) for this class of problems and showed how the approximability of each problem is determined by the corresponding SDP (upto an arbitrarily small additive error) assuming the UGC. He noted that his techniques do no apply to CSPs with strict constraints (all of which must be satisfied) such as Vertex Cover.In this paper we address the approximability of these strict-CSPs. In the class of CSPs we consider, one is given a set of constraints over a set of variables, and a cost function over the assignments, the goal is to find an assignment to the variables of minimum cost which satisfies all the constraints. We present a generic linear program (LP) for a large class of strict-CSPs and give a systematic way to convert integrality gaps for this LP into UGC-based inapproximability results. Some important problems whose approximability our framework captures are Vertex Cover, Hypergraph Vertex Cover, k-partite-Hypergraph Vertex Cover, Independent Set and other covering and packing problems over q-ary alphabets, and a scheduling problem. For the covering and packing problems, which occur quite commonly in practice as well, we provide a matching rounding algorithm, thus settling their approximability upto an arbitrarily small additive error.

We show that for every positive ε > 0, unless N P ⊂ BPQP, it is impossible to approximate the maximum quadratic assignment problem within a factor better than 2 log 1−ε n by a reduction from the maximum label cover problem. Our result also implies that Approximate Graph Isomorphism is not robust and is in fact, 1−ǫ vs ǫ hard assuming the Unique Games Conjecture.Then, we present an O( √ n)-approximation algorithm for the problem based on rounding of the linear programming relaxation often used in the state of the art exact algorithms. * The conference version of the paper appeared at ICALP 2010.