1,405 research outputs found
Clustering of solutions in hard satisfiability problems
We study the structure of the solution space and behavior of local search
methods on random 3-SAT problems close to the SAT/UNSAT transition. Using the
overlap measure of similarity between different solutions found on the same
problem instance we show that the solution space is shrinking as a function of
alpha. We consider chains of satisfiability problems, where clauses are added
sequentially. In each such chain, the overlap distribution is first smooth, and
then develops a tiered structure, indicating that the solutions are found in
well separated clusters. On chains of not too large instances, all solutions
are eventually observed to be in only one small cluster before vanishing. This
condensation transition point is estimated to be alpha_c = 4.26. The transition
approximately obeys finite-size scaling with an apparent critical exponent of
about 1.7. We compare the solutions found by a local heuristic, ASAT, and the
Survey Propagation algorithm up to alpha_c.Comment: 8 pages, 9 figure
Decentralized Constraint Satisfaction
We show that several important resource allocation problems in wireless
networks fit within the common framework of Constraint Satisfaction Problems
(CSPs). Inspired by the requirements of these applications, where variables are
located at distinct network devices that may not be able to communicate but may
interfere, we define natural criteria that a CSP solver must possess in order
to be practical. We term these algorithms decentralized CSP solvers. The best
known CSP solvers were designed for centralized problems and do not meet these
criteria. We introduce a stochastic decentralized CSP solver and prove that it
will find a solution in almost surely finite time, should one exist, also
showing it has many practically desirable properties. We benchmark the
algorithm's performance on a well-studied class of CSPs, random k-SAT,
illustrating that the time the algorithm takes to find a satisfying assignment
is competitive with stochastic centralized solvers on problems with order a
thousand variables despite its decentralized nature. We demonstrate the
solver's practical utility for the problems that motivated its introduction by
using it to find a non-interfering channel allocation for a network formed from
data from downtown Manhattan
Relationship between clustering and algorithmic phase transitions in the random k-XORSAT model and its NP-complete extensions
We study the performances of stochastic heuristic search algorithms on
Uniquely Extendible Constraint Satisfaction Problems with random inputs. We
show that, for any heuristic preserving the Poissonian nature of the underlying
instance, the (heuristic-dependent) largest ratio of constraints per
variables for which a search algorithm is likely to find solutions is smaller
than the critical ratio above which solutions are clustered and
highly correlated. In addition we show that the clustering ratio can be reached
when the number k of variables per constraints goes to infinity by the
so-called Generalized Unit Clause heuristic.Comment: 15 pages, 4 figures, Proceedings of the International Workshop on
Statistical-Mechanical Informatics, September 16-19, 2007, Kyoto, Japan; some
imprecisions in the previous version have been correcte
Biased landscapes for random Constraint Satisfaction Problems
The typical complexity of Constraint Satisfaction Problems (CSPs) can be
investigated by means of random ensembles of instances. The latter exhibit many
threshold phenomena besides their satisfiability phase transition, in
particular a clustering or dynamic phase transition (related to the tree
reconstruction problem) at which their typical solutions shatter into
disconnected components. In this paper we study the evolution of this
phenomenon under a bias that breaks the uniformity among solutions of one CSP
instance, concentrating on the bicoloring of k-uniform random hypergraphs. We
show that for small k the clustering transition can be delayed in this way to
higher density of constraints, and that this strategy has a positive impact on
the performances of Simulated Annealing algorithms. We characterize the modest
gain that can be expected in the large k limit from the simple implementation
of the biasing idea studied here. This paper contains also a contribution of a
more methodological nature, made of a review and extension of the methods to
determine numerically the discontinuous dynamic transition threshold.Comment: 32 pages, 16 figure
Large-scale parallelism for constraint-based local search: the costas array case study
International audienceWe present the parallel implementation of a constraint-based Local Search algorithm and investigate its performance on several hardware plat-forms with several hundreds or thousands of cores. We chose as the basis for these experiments the Adaptive Search method, an efficient sequential Local Search method for Constraint Satisfaction Problems (CSP). After preliminary experiments on some CSPLib benchmarks, we detail the modeling and solving of a hard combinatorial problem related to radar and sonar applications: the Costas Array Problem. Performance evaluation on some classical CSP bench-marks shows that speedups are very good for a few tens of cores, and good up to a few hundreds of cores. However for a hard combinatorial search problem such as the Costas Array Problem, performance evaluation of the sequential version shows results outperforming previous Local Search implementations, while the parallel version shows nearly linear speedups up to 8,192 cores. The proposed parallel scheme is simple and based on independent multi-walks with no communication between processes during search. We also investigated a cooperative multi-walk scheme where processes share simple information, but this scheme does not seem to improve performance
Modelling and solving temporal reasoning as propositional satisfiability
AbstractRepresenting and reasoning about time dependent information is a key research issue in many areas of computer science and artificial intelligence. One of the best known and widely used formalisms for representing interval-based qualitative temporal information is Allen's interval algebra (IA). The fundamental reasoning task in IA is to find a scenario that is consistent with the given information. This problem is in general NP-complete.In this paper, we investigate how an interval-based representation, or IA network, can be encoded into a propositional formula of Boolean variables and/or predicates in decidable theories. Our task is to discover whether satisfying such a formula can be more efficient than finding a consistent scenario for the original problem. There are two basic approaches to modelling an IA network: one represents the relations between intervals as variables and the other represents the end-points of each interval as variables. By combining these two approaches with three different Boolean satisfiability (SAT) encoding schemes, we produced six encoding schemes for converting IA to SAT. In addition, we also showed how IA networks can be formulated into satisfiability modulo theories (SMT) formulae based on the quantifier-free integer difference logic (QF-IDL). These encodings were empirically studied using randomly generated IA problems of sizes ranging from 20 to 100 nodes. A general conclusion we draw from these experimental results is that encoding IA into SAT produces better results than existing approaches. More specifically, we show that the new point-based 1-D support SAT encoding of IA produces consistently better results than the other alternatives considered. In comparison with the six different SAT encodings, the SMT encoding came fourth after the point-based and interval-based 1-D support schemes and the point-based direct scheme. Further, we observe that the phase transition region maps directly from the IA encoding to each SAT or SMT encoding, but, surprisingly, the location of the hard region varies according to the encoding scheme. Our results also show a fixed performance ranking order over the various encoding schemes
The Quantum Adiabatic Algorithm applied to random optimization problems: the quantum spin glass perspective
Among various algorithms designed to exploit the specific properties of
quantum computers with respect to classical ones, the quantum adiabatic
algorithm is a versatile proposition to find the minimal value of an arbitrary
cost function (ground state energy). Random optimization problems provide a
natural testbed to compare its efficiency with that of classical algorithms.
These problems correspond to mean field spin glasses that have been extensively
studied in the classical case. This paper reviews recent analytical works that
extended these studies to incorporate the effect of quantum fluctuations, and
presents also some original results in this direction.Comment: 151 pages, 21 figure
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