5 research outputs found
Two novel evolutionary formulations of the graph coloring problem
We introduce two novel evolutionary formulations of the problem of coloring
the nodes of a graph. The first formulation is based on the relationship that
exists between a graph's chromatic 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 second formulation, unlike the first
one, does not tackle one graph at a time, but rather aims at evolving a
`program' to color all graphs belonging to a class whose members all have the
same number of nodes and other common attributes. The heuristics that result
from these formulations have been tested on some of the Second DIMACS
Implementation Challenge benchmark graphs, and have been found to be
competitive when compared to the several other heuristics that have also been
tested on those graphs.Comment: To appear in Journal of Combinatorial Optimizatio
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 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
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
On a Clique-Based Integer Programming Formulation of Vertex Colouring with Applications in Course Timetabling
Vertex colouring is a well-known problem in combinatorial optimisation, whose
alternative integer programming formulations have recently attracted
considerable attention. This paper briefly surveys seven known formulations of
vertex colouring and introduces a formulation of vertex colouring using a
suitable clique partition of the graph. This formulation is applicable in
timetabling applications, where such a clique partition of the conflict graph
is given implicitly. In contrast with some alternatives, the presented
formulation can also be easily extended to accommodate complex performance
indicators (``soft constraints'') imposed in a number of real-life course
timetabling applications. Its performance depends on the quality of the clique
partition, but encouraging empirical results for the Udine Course Timetabling
problem are reported