5 research outputs found
Coloring Graphs having Few Colorings over Path Decompositions
Lokshtanov, Marx, and Saurabh SODA 2011 proved that there is no
time algorithm for
deciding if an -vertex graph with pathwidth
admits a proper vertex coloring with colors unless the Strong Exponential
Time Hypothesis (SETH) is false. We show here that nevertheless, when
, where is the maximum degree in the
graph , there is a better algorithm, at least when there are few colorings.
We present a Monte Carlo algorithm that given a graph along with a path
decomposition of with pathwidth runs in time, that
distinguishes between -colorable graphs having at most proper
-colorings and non--colorable graphs. We also show how to obtain a
-coloring in the same asymptotic running time. Our algorithm avoids
violating SETH for one since high degree vertices still cost too much and the
mentioned hardness construction uses a lot of them.
We exploit a new variation of the famous Alon--Tarsi theorem that has an
algorithmic advantage over the original form. The original theorem shows a
graph has an orientation with outdegree less than at every vertex, with a
different number of odd and even Eulerian subgraphs only if the graph is
-colorable, but there is no known way of efficiently finding such an
orientation. Our new form shows that if we instead count another difference of
even and odd subgraphs meeting modular degree constraints at every vertex
picked uniformly at random, we have a fair chance of getting a non-zero value
if the graph has few -colorings. Yet every non--colorable graph gives a
zero difference, so a random set of constraints stands a good chance of being
useful for separating the two cases.Comment: Strengthened result from uniquely -colorable graphs to graphs with
few -colorings. Also improved running tim
Coloring Graphs Having Few Colorings Over Path Decompositions
Lokshtanov, Marx, and Saurabh SODA 2011 proved that there is no (k-epsilon)^pw(G)poly(n) time algorithm for deciding if an n-vertex graph G with pathwidth pw admits a proper vertex coloring with k colors unless the Strong Exponential Time Hypothesis (SETH) is false, for any constant epsilon>0. We show here that nevertheless, when k>lfloor Delta/2 rfloor + 1, where Delta is the maximum degree in the graph G, there is a better algorithm, at least when there are few colorings. We present a Monte Carlo algorithm that given a graph G along with a path decomposition of G with pathwidth pw(G) runs in (lfloor Delta/2 rfloor + 1)^pw(G)poly(n)s time, that distinguishes between k-colorable graphs having at most s proper k-colorings and non-k-colorable graphs. We also show how to obtain a k-coloring in the same asymptotic running time. Our algorithm avoids violating SETH for one since high degree vertices still cost too much and the mentioned hardness construction uses a lot of them.
We exploit a new variation of the famous Alon--Tarsi theorem that has an algorithmic advantage over the original form. The original theorem shows a graph has an orientation with outdegree less than k at every vertex, with a different number of odd and even Eulerian subgraphs only if the graph is k-colorable, but there is no known way of efficiently finding such an orientation. Our new form shows that if we instead count another difference of even and odd subgraphs meeting modular degree constraints at every vertex picked uniformly at random, we have a fair chance of getting a non-zero value if the graph has few k-colorings. Yet every non-k-colorable graph gives a zero difference, so a random set of constraints stands a good chance of being useful for separating the two cases
Computing the Chromatic Number Using Graph Decompositions via Matrix Rank
Computing the smallest number such that the vertices of a given graph can
be properly -colored is one of the oldest and most fundamental problems in
combinatorial optimization. The -Coloring problem has been studied
intensively using the framework of parameterized algorithmics, resulting in a
very good understanding of the best-possible algorithms for several
parameterizations based on the structure of the graph. While there is an
abundance of work for parameterizations based on decompositions of the graph by
vertex separators, almost nothing is known about parameterizations based on
edge separators. We fill this gap by studying -Coloring parameterized by
cutwidth, and parameterized by pathwidth in bounded-degree graphs. Our research
uncovers interesting new ways to exploit small edge separators.
We present two algorithms for -Coloring parameterized by cutwidth :
a deterministic one that runs in time , where
is the matrix multiplication constant, and a randomized one with
runtime . In sharp contrast to earlier work, the running time is
independent of . The dependence on cutwidth is optimal: we prove that even
3-Coloring cannot be solved in time assuming the
Strong Exponential Time Hypothesis (SETH). Our algorithms rely on a new rank
bound for a matrix that describes compatible colorings. Combined with a simple
communication protocol for evaluating a product of two polynomials, this also
yields an time randomized algorithm for
-Coloring on graphs of pathwidth and maximum degree . Such a runtime
was first obtained by Bj\"orklund, but only for graphs with few proper
colorings. We also prove that this result is optimal in the sense that no
-time algorithm exists assuming
SETH.Comment: 29 pages. An extended abstract appears in the proceedings of the 26th
Annual European Symposium on Algorithms, ESA 201
Coloring Graphs Having Few Colorings Over Path Decompositions
Lokshtanov, Marx, and Saurabh SODA 2011 proved that there is no (k-epsilon)^pw(G)poly(n) time algorithm for deciding if an n-vertex graph G with pathwidth pw admits a proper vertex coloring with k colors unless the Strong Exponential Time Hypothesis (SETH) is false, for any constant epsilon>0. We show here that nevertheless, when k>lfloor Delta/2 rfloor + 1, where Delta is the maximum degree in the graph G, there is a better algorithm, at least when there are few colorings. We present a Monte Carlo algorithm that given a graph G along with a path decomposition of G with pathwidth pw(G) runs in (lfloor Delta/2 rfloor + 1)^pw(G)poly(n)s time, that distinguishes between k-colorable graphs having at most s proper k-colorings and non-k-colorable graphs. We also show how to obtain a k-coloring in the same asymptotic running time. Our algorithm avoids violating SETH for one since high degree vertices still cost too much and the mentioned hardness construction uses a lot of them. We exploit a new variation of the famous Alon--Tarsi theorem that has an algorithmic advantage over the original form. The original theorem shows a graph has an orientation with outdegree less than k at every vertex, with a different number of odd and even Eulerian subgraphs only if the graph is k-colorable, but there is no known way of efficiently finding such an orientation. Our new form shows that if we instead count another difference of even and odd subgraphs meeting modular degree constraints at every vertex picked uniformly at random, we have a fair chance of getting a non-zero value if the graph has few k-colorings. Yet every non-k-colorable graph gives a zero difference, so a random set of constraints stands a good chance of being useful for separating the two cases