6 research outputs found
On the probabilistic approach to the random satisfiability problem
In this note I will review some of the recent results that have been obtained
in the probabilistic approach to the random satisfiability problem. At the
present moment the results are only heuristic. In the case of the random
3-satisfiability problem a phase transition from the satisfiable to the
unsatisfiable phase is found at . There are other values of
that separates different regimes and they will be described in
details. In this context the properties of the survey decimation algorithm will
also be discussed.Comment: 11 pages, proceedings of SAT 200
Statistical mechanics of optimization problems
Here I will present an introduction to the results that have been recently
obtained in constraint optimization of random problems using statistical
mechanics techniques. After presenting the general results, in order to
simplify the presentation I will describe in details the problems related to
the coloring of a random graph.Comment: proceedings of the conference SigmaPhi di Crete 2005, 10 pages, one
figur
Periodic attractors of random truncator maps
This paper introduces the \textit{truncator} map as a dynamical system on the
space of configurations of an interacting particle system. We represent the
symbolic dynamics generated by this system as a non-commutative algebra and
classify its periodic orbits using properties of endomorphisms of the resulting
algebraic structure. A stochastic model is constructed on these endomorphisms,
which leads to the classification of the distribution of periodic orbits for
random truncator maps. This framework is applied to investigate the periodic
transitions of Bornholdt's spin market model.Comment: 8 pages, presented at APFA
Comparing Beliefs, Surveys and Random Walks
Survey propagation is a powerful technique from statistical physics that has
been applied to solve the 3-SAT problem both in principle and in practice. We
give, using only probability arguments, a common derivation of survey
propagation, belief propagation and several interesting hybrid methods. We then
present numerical experiments which use WSAT (a widely used random-walk based
SAT solver) to quantify the complexity of the 3-SAT formulae as a function of
their parameters, both as randomly generated and after simplification, guided
by survey propagation. Some properties of WSAT which have not previously been
reported make it an ideal tool for this purpose -- its mean cost is
proportional to the number of variables in the formula (at a fixed ratio of
clauses to variables) in the easy-SAT regime and slightly beyond, and its
behavior in the hard-SAT regime appears to reflect the underlying structure of
the solution space that has been predicted by replica symmetry-breaking
arguments. An analysis of the tradeoffs between the various methods of search
for satisfying assignments shows WSAT to be far more powerful that has been
appreciated, and suggests some interesting new directions for practical
algorithm development.Comment: 8 pages, 5 figure
Learning from Survey Propagation: a Neural Network for MAX-E--SAT
Many natural optimization problems are NP-hard, which implies that they are
probably hard to solve exactly in the worst-case. However, it suffices to get
reasonably good solutions for all (or even most) instances in practice. This
paper presents a new algorithm for computing approximate solutions in
for the Maximum Exact 3-Satisfiability (MAX-E--SAT) problem by
using deep learning methodology. This methodology allows us to create a
learning algorithm able to fix Boolean variables by using local information
obtained by the Survey Propagation algorithm. By performing an accurate
analysis, on random CNF instances of the MAX-E--SAT with several Boolean
variables, we show that this new algorithm, avoiding any decimation strategy,
can build assignments better than a random one, even if the convergence of the
messages is not found. Although this algorithm is not competitive with
state-of-the-art Maximum Satisfiability (MAX-SAT) solvers, it can solve
substantially larger and more complicated problems than it ever saw during
training
On Sparse Discretization for Graphical Games
This short paper concerns discretization schemes for representing and
computing approximate Nash equilibria, with emphasis on graphical games, but
briefly touching on normal-form and poly-matrix games. The main technical
contribution is a representation theorem that informally states that to account
for every exact Nash equilibrium using a nearby approximate Nash equilibrium on
a grid over mixed strategies, a uniform discretization size linear on the
inverse of the approximation quality and natural game-representation parameters
suffices. For graphical games, under natural conditions, the discretization is
logarithmic in the game-representation size, a substantial improvement over the
linear dependency previously required. The paper has five other objectives: (1)
given the venue, to highlight the important, but often ignored, role that work
on constraint networks in AI has in simplifying the derivation and analysis of
algorithms for computing approximate Nash equilibria; (2) to summarize the
state-of-the-art on computing approximate Nash equilibria, with emphasis on
relevance to graphical games; (3) to help clarify the distinction between
sparse-discretization and sparse-support techniques; (4) to illustrate and
advocate for the deliberate mathematical simplicity of the formal proof of the
representation theorem; and (5) to list and discuss important open problems,
emphasizing graphical-game generalizations, which the AI community is most
suitable to solve.Comment: 30 pages. Original research note drafted in Dec. 2002 and posted
online Spring'03 (http://www.cis.upenn.
edu/~mkearns/teaching/cgt/revised_approx_bnd.pdf) as part of a course on
computational game theory taught by Prof. Michael Kearns at the University of
Pennsylvania; First major revision sent to WINE'10; Current version sent to
JAIR on April 25, 201