68 research outputs found
Robustly Solvable Constraint Satisfaction Problems
An algorithm for a constraint satisfaction problem is called robust if it
outputs an assignment satisfying at least -fraction of the
constraints given a -satisfiable instance, where
as . 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
Deciding absorption
We characterize absorption in finite idempotent algebras by means of
J\'onsson absorption and cube term blockers. As an application we show that it
is decidable whether a given subset is an absorbing subuniverse of an algebra
given by the tables of its basic operations
Promises Make Finite (Constraint Satisfaction) Problems Infinitary
The fixed template Promise Constraint Satisfaction Problem (PCSP) is a
recently proposed significant generalization of the fixed template CSP, which
includes approximation variants of satisfiability and graph coloring problems.
All the currently known tractable (i.e., solvable in polynomial time) PCSPs
over finite templates can be reduced, in a certain natural way, to tractable
CSPs. However, such CSPs are often over infinite domains. We show that the
infinity is in fact necessary by proving that a specific finite-domain PCSP,
namely (1-in-3-SAT, Not-All-Equal-3-SAT), cannot be naturally reduced to a
tractable finite-domain CSP, unless P=NP
The wonderland of reflections
A fundamental fact for the algebraic theory of constraint satisfaction
problems (CSPs) over a fixed template is that pp-interpretations between at
most countable \omega-categorical relational structures have two algebraic
counterparts for their polymorphism clones: a semantic one via the standard
algebraic operators H, S, P, and a syntactic one via clone homomorphisms
(capturing identities). We provide a similar characterization which
incorporates all relational constructions relevant for CSPs, that is,
homomorphic equivalence and adding singletons to cores in addition to
pp-interpretations. For the semantic part we introduce a new construction,
called reflection, and for the syntactic part we find an appropriate weakening
of clone homomorphisms, called h1 clone homomorphisms (capturing identities of
height 1).
As a consequence, the complexity of the CSP of an at most countable
-categorical structure depends only on the identities of height 1
satisfied in its polymorphism clone as well as the the natural uniformity
thereon. This allows us in turn to formulate a new elegant dichotomy conjecture
for the CSPs of reducts of finitely bounded homogeneous structures.
Finally, we reveal a close connection between h1 clone homomorphisms and the
notion of compatibility with projections used in the study of the lattice of
interpretability types of varieties.Comment: 24 page
Absorbing Subalgebras, Cyclic Terms, and the Constraint Satisfaction Problem
The Algebraic Dichotomy Conjecture states that the Constraint Satisfaction
Problem over a fixed template is solvable in polynomial time if the algebra of
polymorphisms 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 Mar\'oti) in an elementary and
self-contained way
Absorption in Universal Algebra and CSP
The algebraic approach to Constraint Satisfaction Problem led to many developments in both CSP and universal algebra. The notion of absorption was successfully applied on both sides of the connection. This article introduces the concept of absorption, illustrates its use in a number of basic proofs and provides an overview of the most important results obtained by using it
Finitely Tractable Promise Constraint Satisfaction Problems
The Promise Constraint Satisfaction Problem (PCSP) is a generalization of the Constraint Satisfaction Problem (CSP) that includes approximation variants of satisfiability and graph coloring problems. Barto [LICS \u2719] has shown that a specific PCSP, the problem to find a valid Not-All-Equal solution to a 1-in-3-SAT instance, is not finitely tractable in that it can be solved by a trivial reduction to a tractable CSP, but such a CSP is necessarily over an infinite domain (unless P=NP). We initiate a systematic study of this phenomenon by giving a general necessary condition for finite tractability and characterizing finite tractability within a class of templates - the "basic" tractable cases in the dichotomy theorem for symmetric Boolean PCSPs allowing negations by Brakensiek and Guruswami [SODA\u2718]
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