921 research outputs found
New schemes for simplifying binary constraint satisfaction problems
Finding a solution to a Constraint Satisfaction Problem (CSP) is known to be an NP-hard task. This has motivatedthe multitude of works that have been devoted to developing techniques that simplify CSP instances before or duringtheir resolution.The present work proposes rigidly enforced schemes for simplifying binary CSPs that allow the narrowing of valuedomains, either via value merging or via value suppression. The proposed schemes can be viewed as parametrizedgeneralizations of two widely studied CSP simplification techniques, namely, value merging and neighbourhoodsubstitutability. Besides, we show that both schemes may be strengthened in order to allow variable elimination,which may result in more significant simplifications. This work contributes also to the theory of tractable CSPs byidentifying a new tractable class of binary CSP
Solving Functional Constraints by Variable Substitution
Functional constraints and bi-functional constraints are an important
constraint class in Constraint Programming (CP) systems, in particular for
Constraint Logic Programming (CLP) systems. CP systems with finite domain
constraints usually employ CSP-based solvers which use local consistency, for
example, arc consistency. We introduce a new approach which is based instead on
variable substitution. We obtain efficient algorithms for reducing systems
involving functional and bi-functional constraints together with other
non-functional constraints. It also solves globally any CSP where there exists
a variable such that any other variable is reachable from it through a sequence
of functional constraints. Our experiments on random problems show that
variable elimination can significantly improve the efficiency of solving
problems with functional constraints
Optimal Sparsification for Some Binary CSPs Using Low-degree Polynomials
This paper analyzes to what extent it is possible to efficiently reduce the
number of clauses in NP-hard satisfiability problems, without changing the
answer. Upper and lower bounds are established using the concept of
kernelization. Existing results show that if NP is not contained in coNP/poly,
no efficient preprocessing algorithm can reduce n-variable instances of CNF-SAT
with d literals per clause, to equivalent instances with bits for
any e > 0. For the Not-All-Equal SAT problem, a compression to size
exists. We put these results in a common framework by analyzing
the compressibility of binary CSPs. We characterize constraint types based on
the minimum degree of multivariate polynomials whose roots correspond to the
satisfying assignments, obtaining (nearly) matching upper and lower bounds in
several settings. Our lower bounds show that not just the number of
constraints, but also the encoding size of individual constraints plays an
important role. For example, for Exact Satisfiability with unbounded clause
length it is possible to efficiently reduce the number of constraints to n+1,
yet no polynomial-time algorithm can reduce to an equivalent instance with
bits for any e > 0, unless NP is a subset of coNP/poly.Comment: Updated the cross-composition in lemma 18 (minor update), since the
previous version did NOT satisfy requirement 4 of lemma 18 (the proof of
Claim 20 was incorrect
Breaking Instance-Independent Symmetries In Exact Graph Coloring
Code optimization and high level synthesis can be posed as constraint
satisfaction and optimization problems, such as graph coloring used in register
allocation. Graph coloring is also used to model more traditional CSPs relevant
to AI, such as planning, time-tabling and scheduling. Provably optimal
solutions may be desirable for commercial and defense applications.
Additionally, for applications such as register allocation and code
optimization, naturally-occurring instances of graph coloring are often small
and can be solved optimally. A recent wave of improvements in algorithms for
Boolean satisfiability (SAT) and 0-1 Integer Linear Programming (ILP) suggests
generic problem-reduction methods, rather than problem-specific heuristics,
because (1) heuristics may be upset by new constraints, (2) heuristics tend to
ignore structure, and (3) many relevant problems are provably inapproximable.
Problem reductions often lead to highly symmetric SAT instances, and
symmetries are known to slow down SAT solvers. In this work, we compare several
avenues for symmetry breaking, in particular when certain kinds of symmetry are
present in all generated instances. Our focus on reducing CSPs to SAT allows us
to leverage recent dramatic improvement in SAT solvers and automatically
benefit from future progress. We can use a variety of black-box SAT solvers
without modifying their source code because our symmetry-breaking techniques
are static, i.e., we detect symmetries and add symmetry breaking predicates
(SBPs) during pre-processing.
An important result of our work is that among the types of
instance-independent SBPs we studied and their combinations, the simplest and
least complete constructions are the most effective. Our experiments also
clearly indicate that instance-independent symmetries should mostly be
processed together with instance-specific symmetries rather than at the
specification level, contrary to what has been suggested in the literature
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