Beating the Random Ordering Is Hard: Every Ordering CSP Is Approximation Resistant

We prove that, assuming the Unique Games Conjecture (UGC), every problem in the class of ordering constraint satisfaction problems (OCSP) 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 ordering CSP, 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 ⊆ Sk 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 ⊆ Sk, 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. ∗Computer Science Department, Carnegie Mellon University, Pittsburgh, PA 15213. Some of this work was done while visiting the School of Mathematics, Institute for Advanced Study, Princeton, NJ. Research supported in part by a Packard Fellowship, and NSF grants CCF-0343672 and CCF-0963975. Email: [email protected] †School of Computer Science and Communication, KTH, Sweden. Email: [email protected]. Research supported by ERC grant 226203. ‡College of Computing, Georgia Institute of Technology, Atlanta, GA. Some of this work was done while visiting Princeton University and when at Microsoft Research New England, Cambridge, MA. Supported in part by NSF grant CCF-0343672. Email: [email protected] §Department of Computer Science, Princeton University, NJ. Supported by NSF grants MSPA-MCS 0528414, and ITR 0205594. Email: {rajsekar,moses}@cs.princeton.edu ¶Preliminary versions appeared in the Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, 2008 [14] and the 24th IEEE Conference on Computational Complexity, 2009 [6]. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 27 (2011)