5,403 research outputs found
GRASP/VND Optimization Algorithms for Hard Combinatorial Problems
Two hard combinatorial problems are addressed in this thesis. The first one is known as the ”Max CutClique”, a combinatorial problem introduced by P. Martins in 2012. Given a simple graph, the goal is to
find a clique C such that the number of links shared between C and its complement C
C is maximum.
In a first contribution, a GRASP/VND methodology is proposed to tackle the problem. In a second
one, the N P-Completeness of the problem is mathematically proved. Finally, a further generalization
with weighted links is formally presented with a mathematical programming formulation, and the
previous GRASP is adapted to the new problem.
The second problem under study is a celebrated optimization problem coming from network
reliability analysis. We assume a graph G with perfect nodes and imperfect links, that fail independently
with identical probability ρ ∈ [0,1]. The reliability RG(ρ), is the probability that the resulting subgraph
has some spanning tree. Given a number of nodes and links, p and q, the goal is to find the (p,q)-graph
that has the maximum reliability RG(ρ), uniformly in the compact set ρ ∈ [0,1]. In a first contribution,
we exploit properties shared by all uniformly most-reliable graphs such as maximum connectivity and
maximum Kirchhoff number, in order to build a novel GRASP/VND methodology. Our proposal finds
the globally optimum solution under small cases, and it returns novel candidates of uniformly
most-reliable graphs, such as Kantor-Mobius and Heawood graphs. We also offer a literature review, ¨
and a mathematical proof that the bipartite graph K4,4 is uniformly most-reliable.
Finally, an abstract mathematical model of Stochastic Binary Systems (SBS) is also studied. It is a
further generalization of network reliability models, where failures are modelled by a general logical
function. A geometrical approximation of a logical function is offered, as well as a novel method to find
reliability bounds for general SBS. This bounding method combines an algebraic duality, Markov
inequality and Hahn-Banach separation theorem between convex and compact sets
A review of Monte Carlo simulations of polymers with PERM
In this review, we describe applications of the pruned-enriched Rosenbluth
method (PERM), a sequential Monte Carlo algorithm with resampling, to various
problems in polymer physics. PERM produces samples according to any given
prescribed weight distribution, by growing configurations step by step with
controlled bias, and correcting "bad" configurations by "population control".
The latter is implemented, in contrast to other population based algorithms
like e.g. genetic algorithms, by depth-first recursion which avoids storing all
members of the population at the same time in computer memory. The problems we
discuss all concern single polymers (with one exception), but under various
conditions: Homopolymers in good solvents and at the point, semi-stiff
polymers, polymers in confining geometries, stretched polymers undergoing a
forced globule-linear transition, star polymers, bottle brushes, lattice
animals as a model for randomly branched polymers, DNA melting, and finally --
as the only system at low temperatures, lattice heteropolymers as simple models
for protein folding. PERM is for some of these problems the method of choice,
but it can also fail. We discuss how to recognize when a result is reliable,
and we discuss also some types of bias that can be crucial in guiding the
growth into the right directions.Comment: 29 pages, 26 figures, to be published in J. Stat. Phys. (2011
Evolutionary games on graphs
Game theory is one of the key paradigms behind many scientific disciplines
from biology to behavioral sciences to economics. In its evolutionary form and
especially when the interacting agents are linked in a specific social network
the underlying solution concepts and methods are very similar to those applied
in non-equilibrium statistical physics. This review gives a tutorial-type
overview of the field for physicists. The first three sections introduce the
necessary background in classical and evolutionary game theory from the basic
definitions to the most important results. The fourth section surveys the
topological complications implied by non-mean-field-type social network
structures in general. The last three sections discuss in detail the dynamic
behavior of three prominent classes of models: the Prisoner's Dilemma, the
Rock-Scissors-Paper game, and Competing Associations. The major theme of the
review is in what sense and how the graph structure of interactions can modify
and enrich the picture of long term behavioral patterns emerging in
evolutionary games.Comment: Review, final version, 133 pages, 65 figure
A Continuation Method for Nash Equilibria in Structured Games
Structured game representations have recently attracted interest as models
for multi-agent artificial intelligence scenarios, with rational behavior most
commonly characterized by Nash equilibria. This paper presents efficient, exact
algorithms for computing Nash equilibria in structured game representations,
including both graphical games and multi-agent influence diagrams (MAIDs). The
algorithms are derived from a continuation method for normal-form and
extensive-form games due to Govindan and Wilson; they follow a trajectory
through a space of perturbed games and their equilibria, exploiting game
structure through fast computation of the Jacobian of the payoff function. They
are theoretically guaranteed to find at least one equilibrium of the game, and
may find more. Our approach provides the first efficient algorithm for
computing exact equilibria in graphical games with arbitrary topology, and the
first algorithm to exploit fine-grained structural properties of MAIDs.
Experimental results are presented demonstrating the effectiveness of the
algorithms and comparing them to predecessors. The running time of the
graphical game algorithm is similar to, and often better than, the running time
of previous approximate algorithms. The algorithm for MAIDs can effectively
solve games that are much larger than those solvable by previous methods
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