25,971 research outputs found
On the Satisfiability Threshold and Clustering of Solutions of Random 3-SAT Formulas
We study the structure of satisfying assignments of a random 3-SAT formula.
In particular, we show that a random formula of density 4.453 or higher almost
surely has no non-trivial "core" assignments. Core assignments are certain
partial assignments that can be extended to satisfying assignments, and have
been studied recently in connection with the Survey Propagation heuristic for
random SAT. Their existence implies the presence of clusters of solutions, and
they have been shown to exist with high probability below the satisfiability
threshold for k-SAT with k>8, by Achlioptas and Ricci-Tersenghi, STOC 2006. Our
result implies that either this does not hold for 3-SAT or the threshold
density for satisfiability in 3-SAT lies below 4.453.
The main technical tool that we use is a novel simple application of the
first moment method
Single-Step Quantum Search Using Problem Structure
The structure of satisfiability problems is used to improve search algorithms
for quantum computers and reduce their required coherence times by using only a
single coherent evaluation of problem properties. The structure of random k-SAT
allows determining the asymptotic average behavior of these algorithms, showing
they improve on quantum algorithms, such as amplitude amplification, that
ignore detailed problem structure but remain exponential for hard problem
instances. Compared to good classical methods, the algorithm performs better,
on average, for weakly and highly constrained problems but worse for hard
cases. The analytic techniques introduced here also apply to other quantum
algorithms, supplementing the limited evaluation possible with classical
simulations and showing how quantum computing can use ensemble properties of NP
search problems.Comment: 39 pages, 12 figures. Revision describes further improvement with
multiple steps (section 7). See also
http://www.parc.xerox.com/dynamics/www/quantum.htm
On Optimization Modulo Theories, MaxSMT and Sorting Networks
Optimization Modulo Theories (OMT) is an extension of SMT which allows for
finding models that optimize given objectives. (Partial weighted) MaxSMT --or
equivalently OMT with Pseudo-Boolean objective functions, OMT+PB-- is a
very-relevant strict subcase of OMT. We classify existing approaches for MaxSMT
or OMT+PB in two groups: MaxSAT-based approaches exploit the efficiency of
state-of-the-art MAXSAT solvers, but they are specific-purpose and not always
applicable; OMT-based approaches are general-purpose, but they suffer from
intrinsic inefficiencies on MaxSMT/OMT+PB problems.
We identify a major source of such inefficiencies, and we address it by
enhancing OMT by means of bidirectional sorting networks. We implemented this
idea on top of the OptiMathSAT OMT solver. We run an extensive empirical
evaluation on a variety of problems, comparing MaxSAT-based and OMT-based
techniques, with and without sorting networks, implemented on top of
OptiMathSAT and {\nu}Z. The results support the effectiveness of this idea, and
provide interesting insights about the different approaches.Comment: 17 pages, submitted at Tacas 1
Bounded Decentralised Coordination over Multiple Objectives
We propose the bounded multi-objective max-sum algorithm (B-MOMS), the first decentralised coordination algorithm for multi-objective optimisation problems. B-MOMS extends the max-sum message-passing algorithm for decentralised coordination to compute bounded approximate solutions to multi-objective decentralised constraint optimisation problems (MO-DCOPs). Specifically, we prove the optimality of B-MOMS in acyclic constraint graphs, and derive problem dependent bounds on its approximation ratio when these graphs contain cycles. Furthermore, we empirically evaluate its performance on a multi-objective extension of the canonical graph colouring problem. In so doing, we demonstrate that, for the settings we consider, the approximation ratio never exceeds 2, and is typically less than 1.5 for less-constrained graphs. Moreover, the runtime required by B-MOMS on the problem instances we considered never exceeds 30 minutes, even for maximally constrained graphs with agents. Thus, B-MOMS brings the problem of multi-objective optimisation well within the boundaries of the limited capabilities of embedded agents
Hybrid tractability of soft constraint problems
The constraint satisfaction problem (CSP) is a central generic problem in
computer science and artificial intelligence: it provides a common framework
for many theoretical problems as well as for many real-life applications. Soft
constraint problems are a generalisation of the CSP which allow the user to
model optimisation problems. Considerable effort has been made in identifying
properties which ensure tractability in such problems. In this work, we
initiate the study of hybrid tractability of soft constraint problems; that is,
properties which guarantee tractability of the given soft constraint problem,
but which do not depend only on the underlying structure of the instance (such
as being tree-structured) or only on the types of soft constraints in the
instance (such as submodularity). We present several novel hybrid classes of
soft constraint problems, which include a machine scheduling problem,
constraint problems of arbitrary arities with no overlapping nogoods, and the
SoftAllDiff constraint with arbitrary unary soft constraints. An important tool
in our investigation will be the notion of forbidden substructures.Comment: A full version of a CP'10 paper, 26 page
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