23 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 Role of Commutativity in Constraint Propagation Algorithms
Constraint propagation algorithms form an important part of most of the
constraint programming systems. We provide here a simple, yet very general
framework that allows us to explain several constraint propagation algorithms
in a systematic way. In this framework we proceed in two steps. First, we
introduce a generic iteration algorithm on partial orderings and prove its
correctness in an abstract setting. Then we instantiate this algorithm with
specific partial orderings and functions to obtain specific constraint
propagation algorithms.
In particular, using the notions commutativity and semi-commutativity, we
show that the {\tt AC-3}, {\tt PC-2}, {\tt DAC} and {\tt DPC} algorithms for
achieving (directional) arc consistency and (directional) path consistency are
instances of a single generic algorithm. The work reported here extends and
simplifies that of Apt \citeyear{Apt99b}.Comment: 35 pages. To appear in ACM TOPLA
The complexity of global cardinality constraints
In a constraint satisfaction problem (CSP) the goal is to find an assignment
of a given set of variables subject to specified constraints. A global
cardinality constraint is an additional requirement that prescribes how many
variables must be assigned a certain value. We study the complexity of the
problem CCSP(G), the constraint satisfaction problem with global cardinality
constraints that allows only relations from the set G. The main result of this
paper characterizes sets G that give rise to problems solvable in polynomial
time, and states that the remaining such problems are NP-complete
From local to global consistency in temporal constraint networks
AbstractWe study the problem of global consistency for several classes of quantitative temporal constraints which include inequalities, inequations and disjunctions of inequations. In all cases that we consider we identify the level of local consistency that is necessary and sufficient for achieving global consistency and present an algorithm which achieves this level. As a byproduct of our analysis, we also develop an interesting minimal network algorithm
Fundamental properties of neighbourhood substitution in constraint satisfaction problems
AbstractIn combinatorial problems it is often worthwhile simplifying the problem, using operations such as consistency, before embarking on an exhaustive search for solutions. Neighbourhood substitution is such a simplification operation. Whenever a value x for a variable is such that it can be replaced in all constraints by another value y, then x is eliminated.This paper shows that neighbourhood substitutions are important whether the aim is to find one or all solutions. It is proved that the result of a convergent sequence of neighbourhood substitutions is invariant modulo isomorphism. An efficient algorithm is given to find such a sequence. It is also shown that to combine consistency (of any order) and neighbourhood substitution, we only need to establish consistency once
Lower Bounds for Existential Pebble Games and k-Consistency Tests
The existential k-pebble game characterizes the expressive power of the
existential-positive k-variable fragment of first-order logic on finite
structures. The winner of the existential k-pebble game on two given finite
structures can be determined in time O(n2k) by dynamic programming on the graph
of game configurations. We show that there is no O(n(k-3)/12)-time algorithm
that decides which player can win the existential k-pebble game on two given
structures. This lower bound is unconditional and does not rely on any
complexity-theoretic assumptions. Establishing strong k-consistency is a
well-known heuristic for solving the constraint satisfaction problem (CSP). By
the game characterization of Kolaitis and Vardi our result implies that there
is no O(n(k-3)/12)-time algorithm that decides if strong k-consistency can be
established for a given CSP-instance