126 research outputs found
Linear-Time FPT Algorithms via Network Flow
In the area of parameterized complexity, to cope with NP-Hard problems, we
introduce a parameter k besides the input size n, and we aim to design
algorithms (called FPT algorithms) that run in O(f(k)n^d) time for some
function f(k) and constant d. Though FPT algorithms have been successfully
designed for many problems, typically they are not sufficiently fast because of
huge f(k) and d. In this paper, we give FPT algorithms with small f(k) and d
for many important problems including Odd Cycle Transversal and Almost 2-SAT.
More specifically, we can choose f(k) as a single exponential (4^k) and d as
one, that is, linear in the input size. To the best of our knowledge, our
algorithms achieve linear time complexity for the first time for these
problems. To obtain our algorithms for these problems, we consider a large
class of integer programs, called BIP2. Then we show that, in linear time, we
can reduce BIP2 to Vertex Cover Above LP preserving the parameter k, and we can
compute an optimal LP solution for Vertex Cover Above LP using network flow.
Then, we perform an exhaustive search by fixing half-integral values in the
optimal LP solution for Vertex Cover Above LP. A bottleneck here is that we
need to recompute an LP optimal solution after branching. To address this
issue, we exploit network flow to update the optimal LP solution in linear
time.Comment: 20 page
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
Partition a 3-colorable graph into a small bipartite subgraph and a large independent set
Exact algorithms have made a little progress for the 3-coloring problem: improved from to since 1976. The best exact algorithm for the 3-coloring problem is by Beigel and Eppstein, and its analysis is very complicated. We study the parameterized 3-coloring problem: partitioning a 3-colorable graph into a bipartite subgraph and an independent set. Taking the size of the bipartite subgraph as the parameter k, we propose the first parameter algorithm of complexity . Our algorithm can solve the 3-coloring problem faster than the best exact algorithm for graphs with k ≤ 0.527n where n is the graph size. Our study of the parameterized 3-coloring problem brings new insight on studies of the 3-coloring problem. Experiments show that the parameterized algorithm is faster than the exact algorithm for graphs of small parameter k. Moreover, the running time of parameterized algorithm is not much related to edge density, while the running time of exact algorithm increases dramatically as edge density increases
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