37 research outputs found
Generalising tractable VCSPs defined by symmetric tournament pair multimorphisms
We study optimisation problems that can be formulated as valued constraint
satisfaction problems (VCSP). A problem from VCSP is characterised by a
\emph{constraint language}, a fixed set of cost functions taking finite and
infinite costs 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 are interested in \emph{tractable} constraint languages; that is, languages
that give rise to VCSP instances solvable in polynomial time. Cohen et al.
(AIJ'06) have shown that constraint languages that admit the MJN multimorphism
are tractable. Moreover, using a minimisation algorithm for submodular
functions, Cohen et al. (TCS'08) have shown that constraint languages that
admit an STP (symmetric tournament pair) multimorphism are tractable.
We generalise these results by showing that languages admitting the MJN
multimorphism on a subdomain and an STP multimorphisms on the complement of the
subdomain are tractable. The algorithm is a reduction to the algorithm for
languages admitting an STP multimorphism.Comment: 14 page
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 Power of Linear Programming for Valued CSPs
A class of valued constraint satisfaction problems (VCSPs) is characterised
by a valued constraint language, a fixed set of cost functions on a finite
domain. An instance of the problem is specified by a sum of cost functions from
the language with the goal to minimise the sum. This framework includes and
generalises well-studied constraint satisfaction problems (CSPs) and maximum
constraint satisfaction problems (Max-CSPs).
Our main result is a precise algebraic characterisation of valued constraint
languages whose instances can be solved exactly by the basic linear programming
relaxation. Using this result, we obtain tractability of several novel and
previously widely-open classes of VCSPs, including problems over valued
constraint languages that are: (1) submodular on arbitrary lattices; (2)
bisubmodular (also known as k-submodular) on arbitrary finite domains; (3)
weakly (and hence strongly) tree-submodular on arbitrary trees.Comment: Corrected a few typo
The complexity of weighted boolean #CSP*
This paper gives a dichotomy theorem for the complexity of computing the partition
function of an instance of a weighted Boolean constraint satisfaction problem. The problem
is parameterized by a finite set F of nonnegative functions that may be used to assign weights to
the configurations (feasible solutions) of a problem instance. Classical constraint satisfaction problems
correspond to the special case of 0,1-valued functions. We show that computing the partition
function, i.e., the sum of the weights of all configurations, is FP#P-complete unless either (1) every
function in F is of “product type,” or (2) every function in F is “pure affine.” In the remaining cases,
computing the partition function is in P
Relating the Time Complexity of Optimization Problems in Light of the Exponential-Time Hypothesis
Obtaining lower bounds for NP-hard problems has for a long time been an
active area of research. Recent algebraic techniques introduced by Jonsson et
al. (SODA 2013) show that the time complexity of the parameterized SAT()
problem correlates to the lattice of strong partial clones. With this ordering
they isolated a relation such that SAT() can be solved at least as fast
as any other NP-hard SAT() problem. In this paper we extend this method
and show that such languages also exist for the max ones problem
(MaxOnes()) and the Boolean valued constraint satisfaction problem over
finite-valued constraint languages (VCSP()). With the help of these
languages we relate MaxOnes and VCSP to the exponential time hypothesis in
several different ways.Comment: This is an extended version of Relating the Time Complexity of
Optimization Problems in Light of the Exponential-Time Hypothesis, appearing
in Proceedings of the 39th International Symposium on Mathematical
Foundations of Computer Science MFCS 2014 Budapest, August 25-29, 201
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
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