4 research outputs found
A novel evolutionary formulation of the maximum independent set problem
We introduce a novel evolutionary formulation of the problem of finding a
maximum independent set of a graph. The new formulation is based on the
relationship that exists between a graph's independence number and its acyclic
orientations. It views such orientations as individuals and evolves them with
the aid of evolutionary operators that are very heavily based on the structure
of the graph and its acyclic orientations. The resulting heuristic has been
tested on some of the Second DIMACS Implementation Challenge benchmark graphs,
and has been found to be competitive when compared to several of the other
heuristics that have also been tested on those graphs
Network conduciveness with application to the graph-coloring and independent-set optimization transitions
We introduce the notion of a network's conduciveness, a probabilistically
interpretable measure of how the network's structure allows it to be conducive
to roaming agents, in certain conditions, from one portion of the network to
another. We exemplify its use through an application to the two problems in
combinatorial optimization that, given an undirected graph, ask that its
so-called chromatic and independence numbers be found. Though NP-hard, when
solved on sequences of expanding random graphs there appear marked transitions
at which optimal solutions can be obtained substantially more easily than right
before them. We demonstrate that these phenomena can be understood by resorting
to the network that represents the solution space of the problems for each
graph and examining its conduciveness between the non-optimal solutions and the
optimal ones. At the said transitions, this network becomes strikingly more
conducive in the direction of the optimal solutions than it was just before
them, while at the same time becoming less conducive in the opposite direction.
We believe that, besides becoming useful also in other areas in which network
theory has a role to play, network conduciveness may become instrumental in
helping clarify further issues related to NP-hardness that remain poorly
understood
A methodology for determining amino-acid substitution matrices from set covers
We introduce a new methodology for the determination of amino-acid
substitution matrices for use in the alignment of proteins. The new methodology
is based on a pre-existing set cover on the set of residues and on the
undirected graph that describes residue exchangeability given the set cover.
For fixed functional forms indicating how to obtain edge weights from the set
cover and, after that, substitution-matrix elements from weighted distances on
the graph, the resulting substitution matrix can be checked for performance
against some known set of reference alignments and for given gap costs. Finding
the appropriate functional forms and gap costs can then be formulated as an
optimization problem that seeks to maximize the performance of the substitution
matrix on the reference alignment set. We give computational results on the
BAliBASE suite using a genetic algorithm for optimization. Our results indicate
that it is possible to obtain substitution matrices whose performance is either
comparable to or surpasses that of several others, depending on the particular
scenario under consideration
On the phase transitions of graph coloring and independent sets
We study combinatorial indicators related to the characteristic phase
transitions associated with coloring a graph optimally and finding a maximum
independent set. In particular, we investigate the role of the acyclic
orientations of the graph in the hardness of finding the graph's chromatic
number and independence number. We provide empirical evidence that, along a
sequence of increasingly denser random graphs, the fraction of acyclic
orientations that are `shortest' peaks when the chromatic number increases, and
that such maxima tend to coincide with locally easiest instances of the
problem. Similar evidence is provided concerning the `widest' acyclic
orientations and the independence number