665 research outputs found
The algebraic dichotomy conjecture for infinite domain Constraint Satisfaction Problems
We prove that an -categorical core structure primitively positively
interprets all finite structures with parameters if and only if some stabilizer
of its polymorphism clone has a homomorphism to the clone of projections, and
that this happens if and only if its polymorphism clone does not contain
operations , , satisfying the identity .
This establishes an algebraic criterion equivalent to the conjectured
borderline between P and NP-complete CSPs over reducts of finitely bounded
homogenous structures, and accomplishes one of the steps of a proposed strategy
for reducing the infinite domain CSP dichotomy conjecture to the finite case.
Our theorem is also of independent mathematical interest, characterizing a
topological property of any -categorical core structure (the existence
of a continuous homomorphism of a stabilizer of its polymorphism clone to the
projections) in purely algebraic terms (the failure of an identity as above).Comment: 15 page
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
Algebraic Properties of Valued Constraint Satisfaction Problem
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]. We introduce the notions of weighted algebras and varieties
and use the Galois connection due to Cohen et al. [SICOMP 2013] to link VCSP
languages to weighted algebras. We show that the difficulty of VCSP depends
only on the weighted variety generated by the associated weighted algebra.
Paralleling the results for CSPs we exhibit a reduction to cores and rigid
cores which allows us to focus on idempotent weighted varieties. Further, we
propose an analogue of the Algebraic CSP Dichotomy Conjecture; prove the
hardness direction and verify that it agrees with known results for VCSPs on
two-element sets [Cohen et al. 2006], finite-valued VCSPs [Thapper and Zivny
2013] and conservative VCSPs [Kolmogorov and Zivny 2013].Comment: arXiv admin note: text overlap with arXiv:1207.6692 by other author
Complexity Classifications for the Valued Constraint Satisfaction Problem
In a valued constraint satisfaction problem (VCSP), the goal is to find an assignment of values to variables that minimizes a given sum of functions. Each function in the sum depends on a subset of variables, takes values which are rational numbers or infinity, and is chosen from a fixed finite set of functions called a constraint language. We study how the computational complexity of this problem depends on the constraint language. We often consider the case where infinite values are disallowed, and refer to such constraint languages as being finite-valued.
If we consider such finite-valued constraint languages, the case where we allow variables to take two values was classified by Cohen et al., who show that submodular functions essentially give rise to the only tractable case. Non-submodular functions can be used to express the NP-hard Max Cut problem. We consider the case where the variables can take three values, and identify a new infinite set of functions called skew bisubmodular functions which imply tractability. We prove that submodularity with respect to some total order and skew bisubmodularity give rise to the only tractable cases, and in all other cases Max Cut can be expressed. We also show that our characterisation of tractable cases is tight, that is, none of the conditions can be omitted. Thus, our results provide a new dichotomy theorem in constraint satisfaction research. We also negatively answer the question of whether multimorphisms can capture all necessary tractable constraint languages.
We then study the VCSP as a homomorphism problem on digraphs. By adapting a proof designed for CSPs we show that each VCSP with a fixed finite constraint language is equivalent to one where the constraint language consists of one {0,infinity}-valued binary function (i.e. a digraph) and one finite-valued unary function. This latter problem is known as the Minimum Cost Homomorphism Problem for digraphs. We also show that our reduction preserves a number of useful algebraic properties of the constraint language.
Finally, given a finite-valued constraint language, we consider the case where the variables of our VCSP are allowed to take four values. We prove that 1-defect chain multimorphisms, which are required in the four element dichotomy of Min CSP, are a special case of more general fractional polymorphisms we call {a,b}-1-defect fractional polymorphisms. We conclude with a conjecture for the four element case, and some interesting open problems which might lead to a tighter description of tractable finite-valued constraint languages on finite domains of any size
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