2,805 research outputs found
Generalized Majority-Minority Operations are Tractable
Generalized majority-minority (GMM) operations are introduced as a common
generalization of near unanimity operations and Mal'tsev operations on finite
sets. We show that every instance of the constraint satisfaction problem (CSP),
where all constraint relations are invariant under a (fixed) GMM operation, is
solvable in polynomial time. This constitutes one of the largest tractable
cases of the CSP
Aggregation of Votes with Multiple Positions on Each Issue
We consider the problem of aggregating votes cast by a society on a fixed set
of issues, where each member of the society may vote for one of several
positions on each issue, but the combination of votes on the various issues is
restricted to a set of feasible voting patterns. We require the aggregation to
be supportive, i.e. for every issue the corresponding component of
every aggregator on every issue should satisfy . We prove that, in such a set-up, non-dictatorial
aggregation of votes in a society of some size is possible if and only if
either non-dictatorial aggregation is possible in a society of only two members
or a ternary aggregator exists that either on every issue is a majority
operation, i.e. the corresponding component satisfies , or on every issue is a minority operation, i.e.
the corresponding component satisfies We then introduce a notion of uniformly non-dictatorial
aggregator, which is defined to be an aggregator that on every issue, and when
restricted to an arbitrary two-element subset of the votes for that issue,
differs from all projection functions. We first give a characterization of sets
of feasible voting patterns that admit a uniformly non-dictatorial aggregator.
Then making use of Bulatov's dichotomy theorem for conservative constraint
satisfaction problems, we connect social choice theory with combinatorial
complexity by proving that if a set of feasible voting patterns has a
uniformly non-dictatorial aggregator of some arity then the multi-sorted
conservative constraint satisfaction problem on , in the sense introduced by
Bulatov and Jeavons, with each issue representing a sort, is tractable;
otherwise it is NP-complete
Necessary conditions for tractability of valued CSPs
The connection between constraint languages and clone theory has been a
fruitful line of research on the complexity of constraint satisfaction
problems. In a recent result, Cohen et al. [SICOMP'13] have characterised a
Galois connection between valued constraint languages and so-called weighted
clones. In this paper, we study the structure of weighted clones. We extend the
results of Creed and Zivny from [CP'11/SICOMP'13] on types of weightings
necessarily contained in every nontrivial weighted clone. This result has
immediate computational complexity consequences as it provides necessary
conditions for tractability of weighted clones and thus valued constraint
languages. We demonstrate that some of the necessary conditions are also
sufficient for tractability, while others are provably not.Comment: To appear in SIAM Journal on Discrete Mathematics (SIDMA
Relational Width of First-Order Expansions of Homogeneous Graphs with Bounded Strict Width
Solving the algebraic dichotomy conjecture for constraint satisfaction problems over structures first-order definable in countably infinite finitely bounded homogeneous structures requires understanding the applicability of local-consistency methods in this setting. We study the amount of consistency (measured by relational width) needed to solve CSP(?) for first-order expansions ? of countably infinite homogeneous graphs ? := (A; E), which happen all to be finitely bounded. We study our problem for structures ? that additionally have bounded strict width, i.e., for which establishing local consistency of an instance of CSP(?) not only decides if there is a solution but also ensures that every solution may be obtained from a locally consistent instance by greedily assigning values to variables, without backtracking.
Our main result is that the structures ? under consideration have relational width exactly (2, ?_?) where ?_? is the maximal size of a forbidden subgraph of ?, but not smaller than 3. It beats the upper bound: (2 m, 3 m) where m = max(arity(?)+1, ?, 3) and arity(?) is the largest arity of a relation in ?, which follows from a sufficient condition implying bounded relational width given in [Manuel Bodirsky and Antoine Mottet, 2018]. Since ?_? may be arbitrarily large, our result contrasts the collapse of the relational bounded width hierarchy for finite structures ?, whose relational width, if finite, is always at most (2,3)
The Complexity of Quantified Constraint Satisfaction: Collapsibility, Sink Algebras, and the Three-Element Case
The constraint satisfaction probem (CSP) is a well-acknowledged framework in
which many combinatorial search problems can be naturally formulated. The CSP
may be viewed as the problem of deciding the truth of a logical sentence
consisting of a conjunction of constraints, in front of which all variables are
existentially quantified. The quantified constraint satisfaction problem (QCSP)
is the generalization of the CSP where universal quantification is permitted in
addition to existential quantification. The general intractability of these
problems has motivated research studying the complexity of these problems under
a restricted constraint language, which is a set of relations that can be used
to express constraints.
This paper introduces collapsibility, a technique for deriving positive
complexity results on the QCSP. In particular, this technique allows one to
show that, for a particular constraint language, the QCSP reduces to the CSP.
We show that collapsibility applies to three known tractable cases of the QCSP
that were originally studied using disparate proof techniques in different
decades: Quantified 2-SAT (Aspvall, Plass, and Tarjan 1979), Quantified
Horn-SAT (Karpinski, Kleine B\"{u}ning, and Schmitt 1987), and Quantified
Affine-SAT (Creignou, Khanna, and Sudan 2001). This reconciles and reveals
common structure among these cases, which are describable by constraint
languages over a two-element domain. In addition to unifying these known
tractable cases, we study constraint languages over domains of larger size
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
Conservative constraint satisfaction re-revisited
Conservative constraint satisfaction problems (CSPs) constitute an important
particular case of the general CSP, in which the allowed values of each
variable can be restricted in an arbitrary way. Problems of this type are well
studied for graph homomorphisms. A dichotomy theorem characterizing
conservative CSPs solvable in polynomial time and proving that the remaining
ones are NP-complete was proved by Bulatov in 2003. Its proof, however, is
quite long and technical. A shorter proof of this result based on the absorbing
subuniverses technique was suggested by Barto in 2011. In this paper we give a
short elementary prove of the dichotomy theorem for the conservative CSP
Tractability in Constraint Satisfaction Problems: A Survey
International audienceEven though the Constraint Satisfaction Problem (CSP) is NP-complete, many tractable classes of CSP instances have been identified. After discussing different forms and uses of tractability, we describe some landmark tractable classes and survey recent theoretical results. Although we concentrate on the classical CSP, we also cover its important extensions to infinite domains and optimisation, as well as #CSP and QCSP
On tractability and congruence distributivity
Constraint languages that arise from finite algebras have recently been the
object of study, especially in connection with the Dichotomy Conjecture of
Feder and Vardi. An important class of algebras are those that generate
congruence distributive varieties and included among this class are lattices,
and more generally, those algebras that have near-unanimity term operations. An
algebra will generate a congruence distributive variety if and only if it has a
sequence of ternary term operations, called Jonsson terms, that satisfy certain
equations.
We prove that constraint languages consisting of relations that are invariant
under a short sequence of Jonsson terms are tractable by showing that such
languages have bounded relational width
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