8 research outputs found
On the Practical use of Variable Elimination in Constraint Optimization Problems: 'Still-life' as a Case Study
Variable elimination is a general technique for constraint processing. It is
often discarded because of its high space complexity. However, it can be
extremely useful when combined with other techniques. In this paper we study
the applicability of variable elimination to the challenging problem of finding
still-lifes. We illustrate several alternatives: variable elimination as a
stand-alone algorithm, interleaved with search, and as a source of good quality
lower bounds. We show that these techniques are the best known option both
theoretically and empirically. In our experiments we have been able to solve
the n=20 instance, which is far beyond reach with alternative approaches
A Logical Approach to Efficient Max-SAT solving
Weighted Max-SAT is the optimization version of SAT and many important
problems can be naturally encoded as such. Solving weighted Max-SAT is an
important problem from both a theoretical and a practical point of view. In
recent years, there has been considerable interest in finding efficient solving
techniques. Most of this work focus on the computation of good quality lower
bounds to be used within a branch and bound DPLL-like algorithm. Most often,
these lower bounds are described in a procedural way. Because of that, it is
difficult to realize the {\em logic} that is behind.
In this paper we introduce an original framework for Max-SAT that stresses
the parallelism with classical SAT. Then, we extend the two basic SAT solving
techniques: {\em search} and {\em inference}. We show that many algorithmic
{\em tricks} used in state-of-the-art Max-SAT solvers are easily expressable in
{\em logic} terms with our framework in a unified manner.
Besides, we introduce an original search algorithm that performs a restricted
amount of {\em weighted resolution} at each visited node. We empirically
compare our algorithm with a variety of solving alternatives on several
benchmarks. Our experiments, which constitute to the best of our knowledge the
most comprehensive Max-sat evaluation ever reported, show that our algorithm is
generally orders of magnitude faster than any competitor
Cutset Sampling for Bayesian Networks
The paper presents a new sampling methodology for Bayesian networks that
samples only a subset of variables and applies exact inference to the rest.
Cutset sampling is a network structure-exploiting application of the
Rao-Blackwellisation principle to sampling in Bayesian networks. It improves
convergence by exploiting memory-based inference algorithms. It can also be
viewed as an anytime approximation of the exact cutset-conditioning algorithm
developed by Pearl. Cutset sampling can be implemented efficiently when the
sampled variables constitute a loop-cutset of the Bayesian network and, more
generally, when the induced width of the networks graph conditioned on the
observed sampled variables is bounded by a constant w. We demonstrate
empirically the benefit of this scheme on a range of benchmarks
Boosting Search with Variable Elimination in Constraint Optimization and Constraint Satisfaction Problems
There are two main solving schemas for constraint satisfaction and optimization problems: i) search, whose basic step is branching over the values of a variables, and ii) dynamic programming, whose basic step is variable elimination. Variable elimination is time and space exponential in a graph parameter called induced width, which renders the approach infeasible for many problem classes. However, by restricting variable elimination so that only low arity constraints are processed and recorded, it can be e#ectively combined with search, because the elimination of variables may reduce drastically the search tree size. In thi
Boosting search with variable elimination in constraint optimization and constraint satisfaction problems
Abstract. There are two main solving schemas for constraint satisfaction and optimization problems: i) search, whose basic step is branching over the values of a variables, and ii) dynamic programming, whose basic step is variable elimination. Variable elimination is time and space exponential in a graph parameter called induced width, which renders the approach infeasible for many problem classes. However, by restricting variable elimination so that only low arity constraints are processed and recorded, it can be effectively combined with search, because the elimination of variables may reduce drastically the search tree size. In this paper we introduce BE-BB(k), a hybrid general algorithm that combines search and variable elimination. The parameter k controls the tradeoff between the two strategies. The algorithm is space exponential in k. Regarding time, we show that its complexity is bounded by k and a structural parameter from the constraint graph. We provide experimental evidence that the hybrid algorithm can outperform state-of-the-art algorithms in constraint satisfaction, Max-CSP and Weighted CSP. Especially in optimization tasks, the advantage of our approach over plain search can be overwhelming
Multi-objective optimization in graphical models
Many real-life optimization problems are combinatorial, i.e. they concern a choice of the best solution from a finite but exponentially
large set of alternatives. Besides, the solution quality of many of these problems can often be evaluated from several points of view
(a.k.a. criteria). In that case, each criterion may be described by a different objective function. Some important and well-known
multicriteria scenarios are:
路 In investment optimization one wants to minimize risk and maximize benefits.
路 In travel scheduling one wants to minimize time and cost.
路 In circuit design one wants to minimize circuit area, energy consumption and maximize speed.
路 In knapsack problems one wants to minimize load weight and/or volume and maximize its economical value.
The previous examples illustrate that, in many cases, these multiple criteria are incommensurate (i.e., it is difficult or impossible to
combine them into a single criterion) and conflicting (i.e., solutions that are good with respect one criterion are likely to be bad with
respect to another). Taking into account simultaneously the different criteria is not trivial and several notions of optimality have been
proposed. Independently of the chosen notion of optimality, computing optimal solutions represents an important current research
challenge.
Graphical models are a knowledge representation tool widely used in the Artificial Intelligence field. They seem to be specially
suitable for combinatorial problems. Roughly, graphical models are graphs in which nodes represent variables and the (lack of) arcs
represent conditional independence assumptions. In addition to the graph structure, it is necessary to specify its micro-structure
which tells how particular combinations of instantiations of interdependent variables interact. The graphical model framework
provides a unifying way to model a broad spectrum of systems and a collection of general algorithms to efficiently solve them.
In this Thesis we integrate multi-objective optimization problems into the graphical model paradigm and study how algorithmic
techniques developed in the graphical model context can be extended to multi-objective optimization problems. As we show, multiobjective
optimization problems can be formalized as a particular case of graphical models using the semiring-based framework. It
is, to the best of our knowledge, the first time that graphical models in general, and semiring-based problems in particular are used to
model an optimization problem in which the objective function is partially ordered. Moreover, we show that most of the solving
techniques for mono-objective optimization problems can be naturally extended to the multi-objective context. The result of our work
is the mathematical formalization of multi-objective optimization problems and the development of a set of multiobjective solving
algorithms that have been proved to be efficient in a number of benchmarks.Muchos problemas reales de optimizaci贸n son combinatorios, es decir, requieren de la elecci贸n de la mejor soluci贸n (o soluci贸n
贸ptima) dentro de un conjunto finito pero exponencialmente grande de alternativas. Adem谩s, la mejor soluci贸n de muchos de estos
problemas es, a menudo, evaluada desde varios puntos de vista (tambi茅n llamados criterios). Es este caso, cada criterio puede ser
descrito por una funci贸n objetivo. Algunos escenarios multi-objetivo importantes y bien conocidos son los siguientes:
路 En optimizaci贸n de inversiones se pretende minimizar los riesgos y maximizar los beneficios.
路 En la programaci贸n de viajes se quiere reducir el tiempo de viaje y los costes.
路 En el dise帽o de circuitos se quiere reducir al m铆nimo la zona ocupada del circuito, el consumo de energ铆a y maximizar la
velocidad.
路 En los problemas de la mochila se quiere minimizar el peso de la carga y/o el volumen y maximizar su valor econ贸mico.
Los ejemplos anteriores muestran que, en muchos casos, estos criterios son inconmensurables (es decir, es dif铆cil o imposible
combinar todos ellos en un 煤nico criterio) y est谩n en conflicto (es decir, soluciones que son buenas con respecto a un criterio es
probable que sean malas con respecto a otra). Tener en cuenta de forma simult谩nea todos estos criterios no es trivial y para ello se
han propuesto diferentes nociones de optimalidad. Independientemente del concepto de optimalidad elegido, el c贸mputo de
soluciones 贸ptimas representa un importante desaf铆o para la investigaci贸n actual.
Los modelos gr谩ficos son una herramienta para la represetanci贸n del conocimiento ampliamente utilizados en el campo de la
Inteligencia Artificial que parecen especialmente indicados en problemas combinatorios. A grandes rasgos, los modelos gr谩ficos son
grafos en los que los nodos representan variables y la (falta de) arcos representa la interdepencia entre variables. Adem谩s de la
estructura gr谩fica, es necesario especificar su (micro-estructura) que indica c贸mo interact煤an instanciaciones concretas de variables
interdependientes. Los modelos gr谩ficos proporcionan un marco capaz de unificar el modelado de un espectro amplio de sistemas y
un conjunto de algoritmos generales capaces de resolverlos eficientemente.
En esta tesis integramos problemas de optimizaci贸n multi-objetivo en el contexto de los modelos gr谩ficos y estudiamos c贸mo
diversas t茅cnicas algor铆tmicas desarrolladas dentro del marco de los modelos gr谩ficos se pueden extender a problemas de
optimizaci贸n multi-objetivo. Como mostramos, este tipo de problemas se pueden formalizar como un caso particular de modelo
gr谩fico usando el paradigma basado en semi-anillos (SCSP). Desde nuestro conocimiento, 茅sta es la primera vez que los modelos
gr谩ficos en general, y el paradigma basado en semi-anillos en particular, se usan para modelar un problema de optimizaci贸n cuya
funci贸n objetivo est谩 parcialmente ordenada. Adem谩s, mostramos que la mayor铆a de t茅cnicas para resolver problemas monoobjetivo
se pueden extender de forma natural al contexto multi-objetivo. El resultado de nuestro trabajo es la formalizaci贸n
matem谩tica de problemas de optimizaci贸n multi-objetivo y el desarrollo de un conjunto de algoritmos capaces de resolver este tipo
de problemas. Adem谩s, demostramos que estos algoritmos son eficientes en un conjunto determinado de benchmarks