Electronic Colloquium on Computational Complexity, Report No. 27 (2011) 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 ⊆ 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.

我们证明，假设唯一的游戏猜想（UGC），在订购约束满意度问题（OCSP）中的每个问题，每个约束都具有恒定的ARITY，换句话说，如果ρ是ρ是预期的分数，通过随机排序满足的约束，然后获得任何ρ>ρ的ρ近似值，对于最简单的订单CSP，最大的无环子（MAS）问题是UG-HARD。具体而言，对于每个常数ε> 0，以下内容：给定一个有针对的图G，其边缘的分数（1-ε）组成的异步子图，找到一个超过（1/2的）是UG-HARD +ε）的边缘。 MAS问题的任意顺序是研究良好的，并且击败MAS的随机排序是一个基本的开放问题。 ，。约束在π中，v的随机排序有望满足ρ= |π| k！ ρ实际上很强：我们证明，对于每个λ⊆s k，并且很难区分约束的a（1-ε）分数的实例从最多ρ +ε分数可以像π一样，是我们结果的特殊情况。我们的结果（无条件地）暗示MAS的简单半决赛松弛不足以获得更好的近似值。