CSP Dichotomy Conjecture Research Articles

A Proof of the CSP Dichotomy Conjecture

Many natural combinatorial problems can be expressed as constraint satisfaction problems. This class of problems is known to be NP-complete in general, but certain restrictions on the form of the constraints can ensure tractability. The standard way to parameterize interesting subclasses of the constraint satisfaction problem is via finite constraint languages. The main problem is to classify those subclasses that are solvable in polynomial time and those that are NP-complete. It was conjectured that if a constraint language has a weak near-unanimity polymorphism then the corresponding constraint satisfaction problem is tractable; otherwise, it is NP-complete. In the article, we present an algorithm that solves Constraint Satisfaction Problem in polynomial time for constraint languages having a weak near unanimity polymorphism, which proves the remaining part of the conjecture. 1

On the complexity of [formula omitted]-coloring for special oriented trees

For a fixed digraph H, the H-coloring problem is the problem of deciding whether a given input digraph G admits a homomorphism to H. The CSP dichotomy conjecture of Feder and Vardi is equivalent to proving that, for any H, the H-coloring problem is in P or NP-complete. We confirm this dichotomy for a certain class of oriented trees, which we call special trees (generalizing earlier results on special triads and polyads). Moreover, we prove that every tractable special oriented tree has bounded width, i.e., the corresponding H-coloring problem is solvable by local consistency checking. Our proof relies on recent algebraic tools, namely characterization of congruence meet-semidistributivity via pointing operations and absorption theory.

Oriented, 2-edge-colored, and 2-vertex-colored homomorphisms

Two graph parameters are equivalent if, for every graph class, they are either both bounded or both unbounded. We provide a list of graph parameters equivalent to the acyclic chromatic number, which contains in particular the 2-edge-colored chromatic number. Recently, the CSP dichotomy conjecture has been reduced to the case of 2-edge-colored homomorphism and to the case of 2-vertex-colored homomorphism. Both reductions are rather long and follow the reduction to the case of oriented homomorphism in “Graphs and homomorphisms” by Hell and Nešetřil. We give another proof for the case of 2-vertex-colored homomorphism via a simple reduction from the case of 2-edge-colored homomorphism. Finally, we prove that deciding if the 2-edge-colored chromatic number of a 2-edge-colored graph is at most 4 is NP-complete, even if restricted to 2-connected subcubic bipartite planar graphs with arbitrarily large girth.

The Complexity of General-Valued CSPs

An instance of the valued constraint satisfaction problem (VCSP) is given by a finite set of variables, a finite domain of labels, and a sum of functions, each function depending on a subset of the variables. Each function can take finite values specifying costs of assignments of labels to its variables or the infinite value, which indicates an infeasible assignment. The goal is to find an assignment of labels to the variables that minimizes the sum. We study, assuming that P $\ne$ NP, how the complexity of this very general problem depends on the set of functions allowed in the instances, the so-called constraint language. The case when all allowed functions take values in $\{0,\infty\}$ corresponds to ordinary CSPs, where one deals only with the feasibility issue, and there is no optimization. This case is the subject of the algebraic CSP dichotomy conjecture predicting for which constraint languages CSPs are tractable (i.e., solvable in polynomial time) and for which they are NP-hard. The case when all...

CSP DICHOTOMY FOR SPECIAL POLYADS

For a digraph ℍ, the Constraint Satisfaction Problem with template ℍ, or CSP(ℍ), is the problem of deciding whether a given input digraph 𝔾 admits a homomorphism to ℍ. The CSP dichotomy conjecture of Feder and Vardi states that for any digraph ℍ, CSP(ℍ) is either in P or NP-complete. Barto, Kozik, Maróti and Niven (Proc. Amer. Math. Soc.137 (2009) 2921–2934) confirmed the conjecture for a class of oriented trees called special triads. We generalize this result, establishing the dichotomy for a class of oriented trees which we call special polyads. We prove that every tractable special polyad has bounded width and provide the description of special polyads of width 1. We also construct a tractable special polyad which neither has width 1 nor admits any near-unanimity polymorphism.