218 research outputs found
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
Parameterized Approximation Schemes using Graph Widths
Combining the techniques of approximation algorithms and parameterized
complexity has long been considered a promising research area, but relatively
few results are currently known. In this paper we study the parameterized
approximability of a number of problems which are known to be hard to solve
exactly when parameterized by treewidth or clique-width. Our main contribution
is to present a natural randomized rounding technique that extends well-known
ideas and can be used for both of these widths. Applying this very generic
technique we obtain approximation schemes for a number of problems, evading
both polynomial-time inapproximability and parameterized intractability bounds
Recoloring graphs of treewidth 2
Two (proper) colorings of a graph are adjacent if they differ on exactly one
vertex. Jerrum proved that any -coloring of any d-degenerate graph can
be transformed into any other via a sequence of adjacent colorings. A result of
Bonamy et al. ensures that a shortest transformation can have a quadratic
length even for . Bousquet and Perarnau proved that a linear
transformation exists for between -colorings. It is open to determine
if this bound can be reduced. In this note, we prove that it can be reduced for
graphs of treewidth 2, which are 2-degenerate. There exists a linear
transformation between 5-colorings. It completes the picture for graphs of
treewidth 2 since there exist graphs of treewidth 2 such a linear
transformation between 4-colorings does not exist
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