On Maltsev digraphs

The original publication is available at www.springerlink.com Copyright SpringerWe study digraphs preserved by a Maltsev operation, Maltsev digraphs. We show that these digraphs retract either onto a directed path or to the disjoint union of directed cycles, showing that the constraint satisfaction problem for Maltsev digraphs is in logspace, L. (This was observed in [19] using an indirect argument.) We then generalize results in [19] to show that a Maltsev digraph is preserved not only by a majority operation, but by a class of other operations (e.g., minority, Pixley) and obtain a O(V G4)-time algorithm to recognize Maltsev digraphs. We also prove analogous results for digraphs preserved by conservative Maltsev operations which we use to establish that the list homomorphism problem for Maltsev digraphs is in L. We then give a polynomial time characterisation of Maltsev digraphs admitting a conservative 2-semilattice operation. Finally, we give a simple inductive construction of directed acyclic digraphs preserved by a Maltsev operation.Peer reviewe

On Backdoors to Tractable Constraint Languages

International audienceIn 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 polymor-phisms. 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

Algebra and the Complexity of Digraph CSPs: a Survey

We present a brief survey of some of the key results on the interplay between algebraic and graph-theoretic methods in the study of the complexity of digraph-based constraint satisfaction problems

On Maltsev Digraphs

This is an Open Access article, first published by E-CJ on 25 February 2015.We study digraphs preserved by a Maltsev operation: Maltsev digraphs. We show that these digraphs retract either onto a directed path or to the disjoint union of directed cycles, showing in this way that the constraint satisfaction problem for Maltsev digraphs is in logspace, L. We then generalize results from Kazda (2011) to show that a Maltsev digraph is preserved not only by a majority operation, but by a class of other operations (e.g., minority, Pixley) and obtain a O(|VG|4)-time algorithm to recognize Maltsev digraphs. We also prove analogous results for digraphs preserved by conservative Maltsev operations which we use to establish that the list homomorphism problem for Maltsev digraphs is in L. We then give a polynomial time characterisation of Maltsev digraphs admitting a conservative 2-semilattice operation. Finally, we give a simple inductive construction of directed acyclic digraphs preserved by a Maltsev operation, and relate them with series parallel digraphs.Peer reviewedFinal Published versio