26,569 research outputs found
The power of linear programming for general-valued CSPs
Let , called the domain, be a fixed finite set and let , called
the valued constraint language, be a fixed set of functions of the form
, where different functions might have
different arity . We study the valued constraint satisfaction problem
parametrised by , denoted by VCSP. These are minimisation
problems given by variables and the objective function given by a sum of
functions from , each depending on a subset of the variables.
Finite-valued constraint languages contain functions that take on only rational
values and not infinite values.
Our main result is a precise algebraic characterisation of valued constraint
languages whose instances can be solved exactly by the basic linear programming
relaxation (BLP). For a valued constraint language , BLP is a decision
procedure for if and only if admits a symmetric fractional
polymorphism of every arity. For a finite-valued constraint language ,
BLP is a decision procedure if and only if admits a symmetric
fractional polymorphism of some arity, or equivalently, if admits a
symmetric fractional polymorphism of arity 2.
Using these results, we obtain tractability of several novel classes of
problems, including problems over valued constraint languages that are: (1)
submodular on arbitrary lattices; (2) -submodular on arbitrary finite
domains; (3) weakly (and hence strongly) tree-submodular on arbitrary trees.Comment: A full version of a FOCS'12 paper by the last two authors
(arXiv:1204.1079) and an ICALP'13 paper by the first author (arXiv:1207.7213)
to appear in SIAM Journal on Computing (SICOMP
Solving Set Constraint Satisfaction Problems using ROBDDs
In this paper we present a new approach to modeling finite set domain
constraint problems using Reduced Ordered Binary Decision Diagrams (ROBDDs). We
show that it is possible to construct an efficient set domain propagator which
compactly represents many set domains and set constraints using ROBDDs. We
demonstrate that the ROBDD-based approach provides unprecedented flexibility in
modeling constraint satisfaction problems, leading to performance improvements.
We also show that the ROBDD-based modeling approach can be extended to the
modeling of integer and multiset constraint problems in a straightforward
manner. Since domain propagation is not always practical, we also show how to
incorporate less strict consistency notions into the ROBDD framework, such as
set bounds, cardinality bounds and lexicographic bounds consistency. Finally,
we present experimental results that demonstrate the ROBDD-based solver
performs better than various more conventional constraint solvers on several
standard set constraint problems
Rigorous solution techniques for numerical constraint satisfaction problems
A constraint satisfaction problem (e.g., a system of equations and inequalities) consists of a finite set of constraints specifying which value combinations from given variable domains are admitted. It is called numerical if its variable domains are continuous. Such problems arise in many applications, but form a difficult problem class since they are NP-hard. Solving a constraint satisfaction problem is to find one or more value combinations satisfying all its constraints. Numerical computations on floating-point numbers in computers often suffer from rounding errors. The rigorous control of rounding errors during numerical computations is highly desired in many applications because it would benefit the quality and reliability of the decisions based on the solutions found by the computations. Various aspects of rigorous numerical computations in solving constraint satisfaction problems are addressed in this thesis: search, constraint propagation, combination of inclusion techniques, and post-processing. The solution of a constraint satisfaction problem is essentially performed by a search. In this thesis, we propose a new complete search technique (i.e., it can find all solutions within a predetermined tolerance) for numerical constraint satisfaction problems. This technique is general and can be used in place of branching steps in most branch-and-prune methods. Moreover, this new technique speeds up the most recent general search strategy (often by an order of magnitude) and provides a concise representation of solutions. To make a constraint satisfaction problem easier to solve, a major approach, called constraint propagation, in the constraint programming1 field is often used to reduce the variable domains (by discarding redundant value combinations from the domains). Basing on directed acyclic graphs, we propose a new constraint propagation technique and a method for coordinating constraint propagation and search. More importantly, we propose a novel generic scheme for combining multiple inclusion techniques2 in numerical constraint propagation. This scheme allows bringing into the constraint propagation framework the strengths of various techniques coming from different fields. To illustrate the flexibility and efficiency of the generic scheme, we base on this scheme and devise several specific combination strategies for rigorous numerical constraint propagation using interval constraint propagation, interval arithmetic, affine arithmetic, and linear programming. Our experiments show that the new propagation techniques outperform previously available methods by 1 to 4 orders of magnitude or more in speed. We also propose several post-processing techniques for the representation of continuums of solutions. Based on connectedness, they allow grouping each cluster of connected solution subsets into a larger subset, thus allowing getting additional grouping information. Potentially, these techniques enable interval-based solution techniques to be alternatives to bounding-volume techniques in applications such as collision detection and interactive graphics. __________________________________________________ 1 Constraint programming is an approach to programming that relies on both reasoning and computing. 2 An inclusion technique is to include a set of interest into enclosures. It is also called an enclosure technique
Approximation for Maximum Surjective Constraint Satisfaction Problems
Maximum surjective constraint satisfaction problems (Max-Sur-CSPs) are
computational problems where we are given a set of variables denoting values
from a finite domain B and a set of constraints on the variables. A solution to
such a problem is a surjective mapping from the set of variables to B such that
the number of satisfied constraints is maximized. We study the approximation
performance that can be acccchieved by algorithms for these problems, mainly by
investigating their relation with Max-CSPs (which are the corresponding
problems without the surjectivity requirement). Our work gives a complexity
dichotomy for Max-Sur-CSP(B) between PTAS and APX-complete, under the
assumption that there is a complexity dichotomy for Max-CSP(B) between PO and
APX-complete, which has already been proved on the Boolean domain and 3-element
domains
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
Branch-and-Prune Search Strategies for Numerical Constraint Solving
When solving numerical constraints such as nonlinear equations and
inequalities, solvers often exploit pruning techniques, which remove redundant
value combinations from the domains of variables, at pruning steps. To find the
complete solution set, most of these solvers alternate the pruning steps with
branching steps, which split each problem into subproblems. This forms the
so-called branch-and-prune framework, well known among the approaches for
solving numerical constraints. The basic branch-and-prune search strategy that
uses domain bisections in place of the branching steps is called the bisection
search. In general, the bisection search works well in case (i) the solutions
are isolated, but it can be improved further in case (ii) there are continuums
of solutions (this often occurs when inequalities are involved). In this paper,
we propose a new branch-and-prune search strategy along with several variants,
which not only allow yielding better branching decisions in the latter case,
but also work as well as the bisection search does in the former case. These
new search algorithms enable us to employ various pruning techniques in the
construction of inner and outer approximations of the solution set. Our
experiments show that these algorithms speed up the solving process often by
one order of magnitude or more when solving problems with continuums of
solutions, while keeping the same performance as the bisection search when the
solutions are isolated.Comment: 43 pages, 11 figure
Constraint Programming viewed as Rule-based Programming
We study here a natural situation when constraint programming can be entirely
reduced to rule-based programming. To this end we explain first how one can
compute on constraint satisfaction problems using rules represented by simple
first-order formulas. Then we consider constraint satisfaction problems that
are based on predefined, explicitly given constraints. To solve them we first
derive rules from these explicitly given constraints and limit the computation
process to a repeated application of these rules, combined with labeling.We
consider here two types of rules. The first type, that we call equality rules,
leads to a new notion of local consistency, called {\em rule consistency} that
turns out to be weaker than arc consistency for constraints of arbitrary arity
(called hyper-arc consistency in \cite{MS98b}). For Boolean constraints rule
consistency coincides with the closure under the well-known propagation rules
for Boolean constraints. The second type of rules, that we call membership
rules, yields a rule-based characterization of arc consistency. To show
feasibility of this rule-based approach to constraint programming we show how
both types of rules can be automatically generated, as {\tt CHR} rules of
\cite{fruhwirth-constraint-95}. This yields an implementation of this approach
to programming by means of constraint logic programming. We illustrate the
usefulness of this approach to constraint programming by discussing various
examples, including Boolean constraints, two typical examples of many valued
logics, constraints dealing with Waltz's language for describing polyhedral
scenes, and Allen's qualitative approach to temporal logic.Comment: 39 pages. To appear in Theory and Practice of Logic Programming
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