Robust Satisfiability for CSPs

Abstract

The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. An algorithm for a constraint satisfaction problem is called robust if it outputs an assignment satisfying at least a (1−f (ϵ))-fraction of constraints for each (1−ϵ)-satisfiable instance (i.e. such that at most a ϵ-fraction of constraints needs to be removed to make the instance satisfiable), where f (ϵ) → 0 as ϵ → 0. We establish an algebraic framework for analyzing constraint satisfaction problems admitting an efficient robust algorithm with functions f of a given growth rate. We use this framework to derive hardness results. We also describe three classes of problems admitting an efficient robust algorithm such that f is O(1/ log (1/ϵ)), O(ϵ 1/k ) for some k > 1, and O(ϵ), respectively. Finally, we give a complete classification of robust satisfiability with a given f for the Boolean case.

Robust Algorithms with Polynomial Loss for Near-Unanimity CSPs

“…It was stated as an open problem in [23] whether every CSP that admits a robust algorithm with loss O(\varepsi 1/k ) admits one where k is bounded by an absolute constant (that does not depend on D). In the context of the above theorem, the problem can be made more specific: is dependence of k on | D| in this theorem avoidable, or is there a strict hierarchy of possible degrees there?…”

“…Having an NU polymorphism is a sufficient condition for both. Another family of problems CSP(\Gamma ) with bounded pathwidth duality was shown to admit robust algorithms with polynomial loss in [23], where the parameter k depends on the pathwidth duality bound (and appears in the algebraic description of this family). This family includes languages not having an NU polymorphism of any arity; see [13,14].…”

“…If f is a polymorphism of every relation in a constraint language \Gamma , then f is called a polymorphism of \Gamma . Many algorithmic properties of CSP(\Gamma ) depend only on the polymorphisms of \Gamma ; see survey [7], as well as [11,23,32,44].…”

“…Since the dual discriminator is a majority operation, every relation in \Gamma is 2-decomposable. Therefore, it follows, e.g., from Lemma 3.2 in [23], that to prove that CSP(\Gamma ) admits a robust algorithm with loss O( \surd \varepsi ), it suffices to prove this for the case when \Gamma consists of all unary and binary relations preserved by the dual discriminator. Such binary constraints are of one of the four kinds described in section 2.2.…”

“…Horn 3-Sat has bounded width but does not admit a robust algorithm with polynomial loss (unless the UGC fails) [27]. The algebraic condition that separates 3-Lin-p and Horn 3-Sat from the CSPs that can potentially be shown to admit a robust algorithm with polynomial loss is known as SD(\vee ) or omitting types 1, 2, and 5"" [23]; see section 2.2 for the description of SD(\vee ) in terms of polymorphisms. The condition SD(\vee ) is also a necessary condition for the logico-combinatorial property of CSPs called``bounded pathwidth duality"" (which says, roughly, that all unsatisfiable instances can be refuted via local propagation in a linear fashion) and possibly a sufficient condition for it too [44].…”