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(LaTeX) Abstract: |
The {\sc Colorful Motif} problem asks if, given a vertex-colored
graph $G$, there exists a subset $S$ of vertices of $G$ such
that the graph induced by $G$ on $S$ is connected and contains
every color in the graph exactly once. The problem is motivated
by applications in computational biology and is also
well-studied from the theoretical point of view. In particular,
it is known to be NP-complete even on trees of maximum degree
three~[Fellows et al, ICALP 2007]. In their pioneering paper
that introduced the color-coding technique, Alon et al.~[STOC
1995] show, {\em inter alia}, that the problem is FPT on general
graphs. More recently, Cygan et al.~[WG 2010] showed that {\sc
Colorful Motif} is NP-complete on {\em comb graphs}, a special
subclass of the set of trees of maximum degree three. They also
showed that the problem is not likely to admit polynomial
kernels on forests.
We continue the study of the kernelization
complexity of the {\sc Colorful Motif} problem restricted to
simple graph classes. Surprisingly, the infeasibility of
polynomial kernelization persists even when the input is
restricted to comb graphs. We demonstrate this by showing a
simple but novel composition algorithm. Further, we show that
the problem restricted to comb graphs admits polynomially many
polynomial kernels. To our knowledge, there are very few
examples of problems with many polynomial kernels known in the
literature. We also show hardness of polynomial kernelization
for certain variants of the problem on trees; this rules out a
general class of approaches for showing many polynomial kernels
for the problem restricted to trees. Finally, we show that the
problem is unlikely to admit polynomial kernels on another
simple graph class, namely the set of all graphs of diameter
two. As an application of our results, we settle the classical complexity of \cds{} on graphs of diameter two --- specifically, we show that it is \NPC. |
URL for the Abstract: |
http://www.springerlink.com/content/cx67550631830076/ |
Keywords: |
Parameterized Algorithms, Kernel lower bounds, Colorful Motif, Comb graphs |
Copyright Message: |
Copyright Springer-Verlag Berlin Heidelberg 2010. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965,
in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law.
Published in the Proceedings of IPEC 2010, Chennai, India, December 13-15, 2010. Lecture Notes in Computer Science, Volume 6478. The original publication is available at www.springerlink.com : http://www.springerlink.com/content/cx67550631830076/ |
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