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  Beyond Max-cut: Lambda-extendible Properties Parameterized Above the Poljak-Turzík Bound

Mnich, M., Philip, G., Saurabh, S., & Suchy, O. (2012). Beyond Max-cut: Lambda-extendible Properties Parameterized Above the Poljak-Turzík Bound. In D. D'Souza, T. Kavitha, & J. Radhakrishnan (Eds.), IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (pp. 412-423). Wadern: Schloss Dagstuhl. doi:10.4230/LIPIcs.FSTTCS.2012.412.

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Item Permalink: https://hdl.handle.net/11858/00-001M-0000-0014-F755-7 Version Permalink: https://hdl.handle.net/11858/00-001M-0000-0024-AD16-8
Genre: Conference Paper
Latex : Beyond Max-cut: $\lambda$-extendible Properties Parameterized Above the {Poljak-Turz\'ik} Bound

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http://drops.dagstuhl.de/opus/volltexte/2012/3877/ (Any fulltext)
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 Creators:
Mnich, Matthias1, Author
Philip, Geevarghese2, Author           
Saurabh, Saket3, Author
Suchy, Ondrej3, Author
Affiliations:
1Cluster of Excellence Multimodal Computing and Interaction, ou_persistent22              
2Algorithms and Complexity, MPI for Informatics, Max Planck Society, ou_24019              
3External Organizations, ou_persistent22              

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 Abstract: Poljak and Turzík (Discrete Math.\ 1986) introduced the notion of λ-extendible properties of graphs as a generalization of the property of being bipartite. They showed that for any 0<λ<1 and λ-extendible property \Pi, any connected graph G on n vertices and m edges contains a spanning subgraph H\in\Pi with at least λ{}m+\frac1-λ}{2}(n-1) edges. The property of being bipartite is λ-extendible for λ=1/2, and thus the Poljak-Turzík bound generalizes the well-known Edwards-Erd\H{o}s bound for \textsc{Max-Cut}. We define a variant, namely \emph{strong} λ-extendibility, to which the Poljak-Turzík bound applies. For a strongly λ-extendible graph property \Pi, we define the parameterized \textsc{Above Poljak-Turzík (\Pi)} problem as follows: Given a connected graph \(G\) on n vertices and m edges and an integer parameter k, does there exist a spanning subgraph H of G such that H\in\Pi and H has at least λ{}m+\frac{1-λ}{2}(n-1)+k edges? The parameter is k, the surplus over the number of edges guaranteed by the Poljak-Turzík bound. We consider properties \Pi for which the \textsc{Above Poljak-Turzík (\Pi)} problem is fixed-parameter tractable (FPT) on graphs which are O(k) vertices away from being a graph in which each block is a clique. We show that for all such properties, \textsc{Above Poljak-Turzík (\Pi)} is FPT for all 0<λ<1. Our results hold for properties of oriented graphs and graphs with edge labels. Our results generalize the recent result of Crowston et al. (ICALP 2012) on \textsc{Max-Cut} parameterized above the Edwards-Erd\H{o}s bound, and yield \textsf{FPT} algorithms for several graph problems parameterized above lower bounds. For instance, we get that the above-guarantee \textsc{Max q-Colorable Subgraph} problem is \textsf{FPT}. Our results also imply that the parameterized above-guarantee \textsc{Oriented Max Acyclic Digraph} problem is \textsf{FPT, thus solving an open question of Raman and Saurabh (Theor.\ Comput.\ Sci.\ 2006).

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Language(s): eng - English
 Dates: 2012-12-10
 Publication Status: Published online
 Pages: -
 Publishing info: -
 Table of Contents: -
 Rev. Type: -
 Identifiers: Other: Local-ID: 4C146804F0CD924CC1257AD8001AA3B8-MnichPhilipSaurabhSuchy2012a
DOI: 10.4230/LIPIcs.FSTTCS.2012.412
URN: urn:nbn:de:0030-drops-38776
BibTex Citekey: MnichPhilipSaurabhSuchy2012a
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Title: 32nd International Conference on Foundations of Software Technology and Theoretical Computer Science
Place of Event: Hyderabad, India
Start-/End Date: 2012-12-15 - 2012-12-17

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Title: IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science
  Abbreviation : FSTTCS 2012
Source Genre: Proceedings
 Creator(s):
D'Souza, Deepak1, Editor
Kavitha, Telikepalli1, Editor
Radhakrishnan, Jaikumar1, Editor
Affiliations:
1 External Organizations, ou_persistent22            
Publ. Info: Wadern : Schloss Dagstuhl
Pages: - Volume / Issue: - Sequence Number: - Start / End Page: 412 - 423 Identifier: ISBN: 978-3-939897-47-7

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Title: Leibniz International Proceedings in Informatics
  Abbreviation : LIPIcs
Source Genre: Series
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Publ. Info: -
Pages: - Volume / Issue: 18 Sequence Number: - Start / End Page: - Identifier: -