1,973 research outputs found
The complexity of finite-valued CSPs
We study the computational complexity of exact minimisation of
rational-valued discrete functions. Let be a set of rational-valued
functions on a fixed finite domain; such a set is called a finite-valued
constraint language. The valued constraint satisfaction problem,
, is the problem of minimising a function given as
a sum of functions from . We establish a dichotomy theorem with respect
to exact solvability for all finite-valued constraint languages defined on
domains of arbitrary finite size.
We show that every constraint language either admits a binary
symmetric fractional polymorphism in which case the basic linear programming
relaxation solves any instance of exactly, or
satisfies a simple hardness condition that allows for a
polynomial-time reduction from Max-Cut to
The complexity of conservative finite-valued CSPs
We study the complexity of valued constraint satisfaction problems (VCSP). A
problem from VCSP is characterised by a \emph{constraint language}, a fixed set
of cost functions over a finite domain. An instance of the problem is specified
by a sum of cost functions from the language and the goal is to minimise the
sum. We consider the case of so-called \emph{conservative} languages; that is,
languages containing all unary cost functions, thus allowing arbitrary
restrictions on the domains of the variables. This problem has been studied by
Bulatov [LICS'03] for -valued languages (i.e. CSP), by
Cohen~\etal\ (AIJ'06) for Boolean domains, by Deineko et al. (JACM'08) for
-valued cost functions (i.e. Max-CSP), and by Takhanov (STACS'10) for
-valued languages containing all finite-valued unary cost
functions (i.e. Min-Cost-Hom).
We give an elementary proof of a complete complexity classification of
conservative finite-valued languages: we show that every conservative
finite-valued language is either tractable or NP-hard. This is the \emph{first}
dichotomy result for finite-valued VCSPs over non-Boolean domains.Comment: 15 page
The complexity of Boolean surjective general-valued CSPs
Valued constraint satisfaction problems (VCSPs) are discrete optimisation
problems with a -valued objective function given as
a sum of fixed-arity functions. In Boolean surjective VCSPs, variables take on
labels from and an optimal assignment is required to use both
labels from . Examples include the classical global Min-Cut problem in
graphs and the Minimum Distance problem studied in coding theory.
We establish a dichotomy theorem and thus give a complete complexity
classification of Boolean surjective VCSPs with respect to exact solvability.
Our work generalises the dichotomy for -valued constraint
languages (corresponding to surjective decision CSPs) obtained by Creignou and
H\'ebrard. For the maximisation problem of -valued
surjective VCSPs, we also establish a dichotomy theorem with respect to
approximability.
Unlike in the case of Boolean surjective (decision) CSPs, there appears a
novel tractable class of languages that is trivial in the non-surjective
setting. This newly discovered tractable class has an interesting mathematical
structure related to downsets and upsets. Our main contribution is identifying
this class and proving that it lies on the borderline of tractability. A
crucial part of our proof is a polynomial-time algorithm for enumerating all
near-optimal solutions to a generalised Min-Cut problem, which might be of
independent interest.Comment: v5: small corrections and improved presentatio
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 ≠ 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, ∞} 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 NP-hard. The case when all allowed functions take only finite values corresponds to finite-valued CSP, where the feasibility aspect is trivial and one deals only with the optimization issue. The complexity of finite-valued CSPs was fully classified by Thapper and Zivny. An algebraic necessary condition for tractability of a general-valued CSP with a fixed constraint language was recently given by Kozik and Ochremiak. As our main result, we prove that if a constraint language satisfies this algebraic necessary condition, and the feasibility CSP (i.e. the problem of deciding whether a given instance has a feasible solution) corresponding to the VCSP with this language is tractable, then the VCSP is tractable. The algorithm is a simple combination of the assumed algorithm for the feasibility CSP and the standard LP relaxation. As a corollary, we obtain that a dichotomy for ordinary CSPs would imply a dichotomy for general-valued CSPs
A Galois Connection for Weighted (Relational) Clones of Infinite Size
A Galois connection between clones and relational clones on a fixed finite
domain is one of the cornerstones of the so-called algebraic approach to the
computational complexity of non-uniform Constraint Satisfaction Problems
(CSPs). Cohen et al. established a Galois connection between finitely-generated
weighted clones and finitely-generated weighted relational clones [SICOMP'13],
and asked whether this connection holds in general. We answer this question in
the affirmative for weighted (relational) clones with real weights and show
that the complexity of the corresponding valued CSPs is preserved
The complexity of general-valued CSPs seen from the other side
The constraint satisfaction problem (CSP) is concerned with homomorphisms
between two structures. For CSPs with restricted left-hand side structures, the
results of Dalmau, Kolaitis, and Vardi [CP'02], Grohe [FOCS'03/JACM'07], and
Atserias, Bulatov, and Dalmau [ICALP'07] establish the precise borderline of
polynomial-time solvability (subject to complexity-theoretic assumptions) and
of solvability by bounded-consistency algorithms (unconditionally) as bounded
treewidth modulo homomorphic equivalence.
The general-valued constraint satisfaction problem (VCSP) is a generalisation
of the CSP concerned with homomorphisms between two valued structures. For
VCSPs with restricted left-hand side valued structures, we establish the
precise borderline of polynomial-time solvability (subject to
complexity-theoretic assumptions) and of solvability by the -th level of the
Sherali-Adams LP hierarchy (unconditionally). We also obtain results on related
problems concerned with finding a solution and recognising the tractable cases;
the latter has an application in database theory.Comment: v2: Full version of a FOCS'18 paper; improved presentation and small
correction
The power of Sherali-Adams relaxations for general-valued CSPs
We give a precise algebraic characterisation of the power of Sherali-Adams
relaxations for solvability of valued constraint satisfaction problems to
optimality. The condition is that of bounded width which has already been shown
to capture the power of local consistency methods for decision CSPs and the
power of semidefinite programming for robust approximation of CSPs.
Our characterisation has several algorithmic and complexity consequences. On
the algorithmic side, we show that several novel and many known valued
constraint languages are tractable via the third level of the Sherali-Adams
relaxation. For the known languages, this is a significantly simpler algorithm
than the previously obtained ones. On the complexity side, we obtain a
dichotomy theorem for valued constraint languages that can express an injective
unary function. This implies a simple proof of the dichotomy theorem for
conservative valued constraint languages established by Kolmogorov and Zivny
[JACM'13], and also a dichotomy theorem for the exact solvability of
Minimum-Solution problems. These are generalisations of Minimum-Ones problems
to arbitrary finite domains. Our result improves on several previous
classifications by Khanna et al. [SICOMP'00], Jonsson et al. [SICOMP'08], and
Uppman [ICALP'13].Comment: Full version of an ICALP'15 paper (arXiv:1502.05301
Binarisation for Valued Constraint Satisfaction Problems
We study methods for transforming valued constraint satisfaction problems (VCSPs) to binary VCSPs. First, we show that the standard dual encoding preserves many aspects of the algebraic properties that capture the computational complexity of VCSPs. Second, we extend the reduction of CSPs to binary CSPs described by Bul´ın et al. [Log. Methods Comput. Sci., 11 (2015)] to VCSPs. This reduction establishes that VCSPs over a fixed valued constraint language are polynomial-time equivalent to minimum-cost homomorphism problems over a fixed digraph
Computational Complexity of the Minimum Cost Homomorphism Problem on Three-Element Domains
In this paper we study the computational complexity of the (extended) minimum
cost homomorphism problem (Min-Cost-Hom) as a function of a constraint
language, i.e. a set of constraint relations and cost functions that are
allowed to appear in instances. A wide range of natural combinatorial
optimisation problems can be expressed as Min-Cost-Homs and a classification of
their complexity would be highly desirable, both from a direct, applied point
of view as well as from a theoretical perspective.
Min-Cost-Hom can be understood either as a flexible optimisation version of
the constraint satisfaction problem (CSP) or a restriction of the
(general-valued) valued constraint satisfaction problem (VCSP). Other
optimisation versions of CSPs such as the minimum solution problem (Min-Sol)
and the minimum ones problem (Min-Ones) are special cases of Min-Cost-Hom.
The study of VCSPs has recently seen remarkable progress. A complete
classification for the complexity of finite-valued languages on arbitrary
finite domains has been obtained Thapper and Zivny [STOC'13]. However,
understanding the complexity of languages that are not finite-valued appears to
be more difficult. Min-Cost-Hom allows us to study problematic languages of
this type without having to deal with with the full generality of the VCSP. A
recent classification for the complexity of three-element Min-Sol, Uppman
[ICALP'13], takes a step in this direction. In this paper we extend this result
considerably by determining the complexity of three-element Min-Cost-Hom
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