1,565 research outputs found
Extremal Optimization at the Phase Transition of the 3-Coloring Problem
We investigate the phase transition of the 3-coloring problem on random
graphs, using the extremal optimization heuristic. 3-coloring is among the
hardest combinatorial optimization problems and is closely related to a 3-state
anti-ferromagnetic Potts model. Like many other such optimization problems, it
has been shown to exhibit a phase transition in its ground state behavior under
variation of a system parameter: the graph's mean vertex degree. This phase
transition is often associated with the instances of highest complexity. We use
extremal optimization to measure the ground state cost and the ``backbone'', an
order parameter related to ground state overlap, averaged over a large number
of instances near the transition for random graphs of size up to 512. For
graphs up to this size, benchmarks show that extremal optimization reaches
ground states and explores a sufficient number of them to give the correct
backbone value after about update steps. Finite size scaling gives
a critical mean degree value . Furthermore, the
exploration of the degenerate ground states indicates that the backbone order
parameter, measuring the constrainedness of the problem, exhibits a first-order
phase transition.Comment: RevTex4, 8 pages, 4 postscript figures, related information available
at http://www.physics.emory.edu/faculty/boettcher
Lower bounds for on-line graph colorings
We propose two strategies for Presenter in on-line graph coloring games. The
first one constructs bipartite graphs and forces any on-line coloring algorithm
to use colors, where is the number of vertices in the
constructed graph. This is best possible up to an additive constant. The second
strategy constructs graphs that contain neither nor as a subgraph
and forces colors. The best known
on-line coloring algorithm for these graphs uses colors
Complexity of Grundy coloring and its variants
The Grundy number of a graph is the maximum number of colors used by the
greedy coloring algorithm over all vertex orderings. In this paper, we study
the computational complexity of GRUNDY COLORING, the problem of determining
whether a given graph has Grundy number at least . We also study the
variants WEAK GRUNDY COLORING (where the coloring is not necessarily proper)
and CONNECTED GRUNDY COLORING (where at each step of the greedy coloring
algorithm, the subgraph induced by the colored vertices must be connected).
We show that GRUNDY COLORING can be solved in time and WEAK
GRUNDY COLORING in time on graphs of order . While GRUNDY
COLORING and WEAK GRUNDY COLORING are known to be solvable in time
for graphs of treewidth (where is the number of
colors), we prove that under the Exponential Time Hypothesis (ETH), they cannot
be solved in time . We also describe an
algorithm for WEAK GRUNDY COLORING, which is therefore
\fpt for the parameter . Moreover, under the ETH, we prove that such a
running time is essentially optimal (this lower bound also holds for GRUNDY
COLORING). Although we do not know whether GRUNDY COLORING is in \fpt, we
show that this is the case for graphs belonging to a number of standard graph
classes including chordal graphs, claw-free graphs, and graphs excluding a
fixed minor. We also describe a quasi-polynomial time algorithm for GRUNDY
COLORING and WEAK GRUNDY COLORING on apex-minor graphs. In stark contrast with
the two other problems, we show that CONNECTED GRUNDY COLORING is
\np-complete already for colors.Comment: 24 pages, 7 figures. This version contains some new results and
improvements. A short paper based on version v2 appeared in COCOON'1
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