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  On the Kernelization Complexity of Colorful Motifs

Ambalath, A. M., Balasundaram, R., H., C. R., Koppula, V., Misra, N., Philip, G., et al. (2010). On the Kernelization Complexity of Colorful Motifs. In V. Raman, & S. Saurabh (Eds.), Parameterized and Exact Computation (pp. 14-25). Berlin: Springer. doi:10.1007/978-3-642-17493-3_4.

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Item Permalink: https://hdl.handle.net/11858/00-001M-0000-0019-DC1D-3 Version Permalink: https://hdl.handle.net/11858/00-001M-0000-0019-DC1E-1
Genre: Conference Paper

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 Creators:
Ambalath, Abhimanyu M.1, Author
Balasundaram, Radheshyam1, Author
H., Chintan Rao1, Author
Koppula, Venkata1, Author
Misra, Neeldhara1, Author
Philip, Geevarghese1, Author           
Ramanujan, M. S.1, Author
Affiliations:
1External Organizations, ou_persistent22              

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 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.

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Language(s): eng - English
 Dates: 20102010
 Publication Status: Issued
 Pages: -
 Publishing info: -
 Table of Contents: -
 Rev. Type: -
 Identifiers: DOI: 10.1007/978-3-642-17493-3_4
BibTex Citekey: AmbalathBalasundaramHKoppulaMisraPhilipRamanujan2010
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Title: IPEC 2010
Place of Event: Chennai, India
Start-/End Date: 2010-12-13 - 2010-12-15

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Title: Parameterized and Exact Computation
  Abbreviation : IPEC 2010
  Subtitle : 5th International Symposium, IPEC 2010, Chennai, India, December 13-15, 2010. Proceedings
Source Genre: Proceedings
 Creator(s):
Raman, Venkatesh1, Editor
Saurabh, Saket1, Editor
Affiliations:
1 External Organizations, ou_persistent22            
Publ. Info: Berlin : Springer
Pages: - Volume / Issue: - Sequence Number: - Start / End Page: 14 - 25 Identifier: ISBN: 978-3-642-17492-6

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Title: Lecture Notes in Computer Science
  Abbreviation : LNCS
Source Genre: Series
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Pages: - Volume / Issue: 6478 Sequence Number: - Start / End Page: - Identifier: -