Todd Niven

The CSP Dichotomy Holds for Digraphs with No Sources and No Sinks (A Positive Answer to a Conjecture of Bang-Jensen and Hell)

Bang-Jensen and Hell conjectured in 1990 (using the language of graph homomorphisms) a constraint satisfaction problem (CSP) dichotomy for digraphs with no sources or sinks. The conjecture states that the CSP for such a digraph is tractable if each component of its core is a cycle and is

Graphs, polymorphisms and the complexity of homomorphism problems

NP

CSP dichotomy for special triads

-complete otherwise. In this paper we prove this conjecture and, as a consequence, a conjecture of Bang-Jensen, Hell, and MacGillivray from 1995 classifying hereditarily hard digraphs. Further, we show that the CSP dichotomy for digraphs with no sources or sinks agrees with the algebraic characterization conjectured by Bulatov, Jeavons, and Krokhin in 2005.

On the reduction of the CSP dichotomy conjecture to digraphs

We use a connection between polymorphisms and the structure of smooth digraphs to prove the conjecture of Bang-Jensen and Hell from 1990 and, as a consequence, a conjecture of Bang-Jensen, Hell and MacGillivray from 1995. The conjectured characterization of computationally complex coloring problems for smooth digraphs is proved using tools of universal algebra. We cite further graph results obtained using this new approach. The proofs are based in an universal algebraic framework developed for the Constraint Satisfaction Problem and the CSP dichotomy conjecture of Feder and Vardi in particular.

A finer reduction of constraint problems to digraphs

For a fixed digraph G, the Constraint Satisfaction Problem with the template G, or CSP(G) for short, is the problem of deciding whether a given input digraph H admits a homomorphism to G. The dichotomy conjecture of Feder and Vardi states that CSP(G), for any choice of G, is solvable in polynomial time or NP-complete. This paper confirms the conjecture for a class of oriented trees called special triads. As a corollary we get the smallest known example of an oriented tree (with 33 vertices) defining an NP-complete CSP(G).

On Maltsev digraphs

It is well known that the constraint satisfaction problem over general relational structures can be reduced in polynomial time to digraphs. We present a simple variant of such a reduction and use it to show that the algebraic dichotomy conjecture is equivalent to its restriction to digraphs and that the polynomial reduction can be made in logspace. We also show that our reduction preserves the bounded width property, i.e., solvability by local consistency methods. We discuss further algorithmic properties that are preserved and related open problems. The first author was supported by the grant projects GACR 201/09/H012, GA UK 67410, SVV-2013-267317; the second author gratefully acknowledges support by the Natural Sciences and Engineering Research Council of Canada in the form of a Discovery Grant; the third and fourth were supported by ARC Discovery Project DP1094578; the first and fourth authors were also supported by the Fields Institute.

Dualizable but not fully dualizable algebras.

It is well known that the constraint satisfaction problem over a general relational structure A is polynomial time equivalent to the constraint problem over some associated digraph. We present a variant of this construction and show that the corresponding constraint satisfaction problem is logspace equivalent to that over A. Moreover, we show that almost all of the commonly encountered polymorphism properties are held equivalently on the A and the constructed digraph. As a consequence, the Algebraic CSP dichotomy conjecture as well as the conjectures characterizing CSPs solvable in logspace and in nondeterministic logspace are equivalent to their restriction to digraphs.

On Maltsev Digraphs

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 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(VG4)-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.

Digraph related constructions and the complexity of digraph homomorphism problems

We give an infinite family of finite dualizable unary algebras that are not fully κ-dualizable, for all cardinals κ.