6,569 research outputs found
On Feedback Vertex Set: New Measure and New Structures
We present a new parameterized algorithm for the {feedback vertex set}
problem ({\sc fvs}) on undirected graphs. We approach the problem by
considering a variation of it, the {disjoint feedback vertex set} problem ({\sc
disjoint-fvs}), which finds a feedback vertex set of size that has no
overlap with a given feedback vertex set of the graph . We develop an
improved kernelization algorithm for {\sc disjoint-fvs} and show that {\sc
disjoint-fvs} can be solved in polynomial time when all vertices in have degrees upper bounded by three. We then propose a new
branch-and-search process on {\sc disjoint-fvs}, and introduce a new
branch-and-search measure. The process effectively reduces a given graph to a
graph on which {\sc disjoint-fvs} becomes polynomial-time solvable, and the new
measure more accurately evaluates the efficiency of the process. These
algorithmic and combinatorial studies enable us to develop an
-time parameterized algorithm for the general {\sc fvs} problem,
improving all previous algorithms for the problem.Comment: Final version, to appear in Algorithmic
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
- …