On Backdoors To Tractable Constraint Languages

TL;DR

This paper investigates the parameterized complexity of detecting small backdoors in constraint satisfaction problems, identifying conditions under which the problem is fixed-parameter tractable or unlikely to be so.

Contribution

It provides a systematic complexity analysis of backdoor detection in CSPs based on polymorphisms, highlighting cases where the problem is FPT and where it is not.

Findings

Detection is unlikely FPT when parameters are r or k alone.

Detection becomes FPT when combining parameters k and r for certain classes.

Identifies classes of languages with tractable backdoor detection.

Abstract

In the context of CSPs, a strong backdoor is a subset of variables such that every complete assignment yields a residual instance guaranteed to have a specified property. If the property allows efficient solving, then a small strong backdoor provides a reasonable decomposition of the original instance into easy instances. An important challenge is the design of algorithms that can find quickly a small strong backdoor if one exists. We present a systematic study of the parameterized complexity of backdoor detection when the target property is a restricted type of constraint language defined by means of a family of polymorphisms. In particular, we show that under the weak assumption that the polymorphisms are idempotent, the problem is unlikely to be FPT when the parameter is either r (the constraint arity) or k (the size of the backdoor) unless P = NP or FPT = W[2]. When the parameter is k+r, however, we are able to identify large classes of languages for which the problem of finding a small backdoor is FPT.