Marcin Kozik

Constraint Satisfaction Problems of Bounded Width

We provide a full characterization of applicability of The Local Consistency Checking algorithm to solving the non-uniform Constraint Satisfaction Problems. This settles the conjecture of Larose and Zadori.

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

Constraint Satisfaction Problems Solvable by Local Consistency Methods

NP

ABSORBING SUBALGEBRAS, CYCLIC TERMS, AND THE CONSTRAINT SATISFACTION PROBLEM ∗

-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.

Robust satisfiability of constraint satisfaction problems

We prove that constraint satisfaction problems without the ability to count are solvable by the local consistency checking algorithm. This settles three (equivalent) conjectures: Feder--Vardi [SICOMP’98], Bulatov [LICS’04] and Larose--Zádori [AU’07].

Graphs, polymorphisms and the complexity of homomorphism problems

The Algebraic Dichotomy Conjecture states that the Constraint Satisfaction Problem over a fixed template is solvable in polynomial time if the algebra of polymor- phisms associated to the template lies in a Taylor variety, and is NP-complete otherwise. This paper provides two new characterizations of finitely generated Taylor varieties. The first characterization is using absorbing subalgebras and the second one cyclic terms. These new conditions allow us to reprove the conjecture of Bang-Jensen and Hell (proved by the authors) and the characterization of locally finite Taylor varieties using weak near- unanimity terms (proved by McKenzie and Maroti) in an elementary and self-contained way.

Algebraic Properties of Valued Constraint Satisfaction Problem

An algorithm for a constraint satisfaction problem is called robust if it outputs an assignment satisfying at least (1-g(ε))-fraction of the constraints given a (1-ε)-satisfiable instance, where g(ε) -> 0 as ε -> 0,

CSP dichotomy for special triads

g(0)=0. Guruswami and Zhou conjectured a characterization of constraint languages for which the corresponding constraint satisfaction problem admits an efficient robust algorithm. This paper confirms their conjecture.

Congruence Distributivity Implies Bounded Width

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.

Near Unanimity Constraints Have Bounded Pathwidth Duality

The paper presents an algebraic framework for optimization problems expressible as Valued Constraint Satisfaction Problems. Our results generalize the algebraic framework for the decision version (CSPs) provided by Bulatov et al. [SICOMP 2005].