On Backdoors to Tractable Constraint Languages

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.

Backdoors into heterogeneous classes of SAT and CSP

“…Recently, two papers dealing with strong backdoor set detection for the constraint satisfaction problem have appeared [7,17], which nicely supplement (however are mostly orthogonal to) the results in this paper. Carbonnel et al [7] show W [2]-hardness for strong backdoor set detection parameterized by the size of the backdoor set even for CSP-instances with only one constraint (however with unbounded domain and unbounded arity).…”

“…Recently, two papers dealing with strong backdoor set detection for the constraint satisfaction problem have appeared [7,17], which nicely supplement (however are mostly orthogonal to) the results in this paper. Carbonnel et al [7] show W [2]-hardness for strong backdoor set detection parameterized by the size of the backdoor set even for CSP-instances with only one constraint (however with unbounded domain and unbounded arity). They also give a fixed-parameter algorithm for strong backdoor set detection parameterized by the size of the backdoor set and the maximum arity of any constraint, if the base class is "h-Helly" for any fixed integer h. However, as is also mentioned in Carbonnel and Cooper [6], the "h-Helly" property is rather restrictive and orthogonal to our approach of considering the domain as an additional parameter.…”

Backdoors to Tractable Valued CSP

“…Gaspers et al [15] considered heterogeneous backdoors into tractable CSP classes that are characterized by polymorphisms. A similar approach was also undertaken by Carbonnel et al [3] who also considered base classes that are "h-Helly" for a fixed integer h under the additional assumption that the domain is a finite subset of the natural numbers and comes with a fixed ordering.…”

Combining Treewidth and Backdoors for CSP

“…A natural way of defining such a class is to consider all CSP instances whose constraints use relations from a constraint language Γ, denoted by CSP(Γ). The last couple of years have seen several new results for CSP using this backdoor-based approach (see, e.g., [12,28,29]. In particular, the general aim of research in this direction is to obtain so-called fixed-parameter algorithms, i.e., algorithms where the running time only has a polynomial dependence on the input size and the exponential blow-up is restricted exclusively to the size of the backdoor (the parameter).…”