Solving minones-2-sat as Fast as vertex cover
The problem of finding a satisfying assignment for a 2-SAT formula that minimizes the number of variables that are set to 1 (min ones 2–sat) is NP-complete. It generalizes the well-studied problem of finding the smallest vertex cover of a graph, which can be modeled using a 2-SAT formula with no negative literals. The natural parameterized version of the problem asks for a satisfying assignment of weight at most k.
In this paper, we present a polynomial-time reduction from min ones 2–sat to vertex cover without increasing the parameter and ensuring that the number of vertices in the reduced instance is equal to the number of variables of the input formula. Consequently, we conclude that this problem also has a simple 2-approximation algorithm and a 2k variables kernel subsuming these results known earlier. Further, the problem admits algorithms for the parameterized and optimization versions whose runtimes will always match the runtimes of the best-known algorithms for the corresponding versions of vertex cover.
Unable to display preview. Download preview PDF.
- [BJG08]Bang-Jensen, J., Gutin, G.Z.: Digraphs: Theory, algorithms and applications. Springer Publishing Company, Heidelberg (2008) (incorporated)Google Scholar
- [DF99]Downey, R.G., Fellows, M.R.: Parameterized complexity. Springer, Heidelberg (November 1999)Google Scholar
- [DJ02]Dahllöf, V., Jonsson, P.: An algorithm for counting maximum weighted independent sets and its applications. In: SODA, pp. 292–298 (2002)Google Scholar
- [Hoc97]Hochbaum, D.S. (ed.): Approximation algorithms for NP-hard problems. PWS Publishing Co., Boston (1997)Google Scholar
- [KLR09]Kneis, J., Langer, A., Rossmanith, P.: A fine-grained analysis of a simple independent set algorithm. In: Kannan, R., Kumar, N. (eds.) IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2009), Leibniz International Proceedings in Informatics (LIPIcs), Dagstuhl, Germany, vol. 4, pp. 287–298. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik (2009)Google Scholar
- [KW09]Kratsch, S., Wahlström, M.: Two edge modification problems without polynomial kernels. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 264–275. Springer, Heidelberg (2009)Google Scholar
- [KWar]Kratsch, S., Wahlström, M.: Preprocessing of min ones problems: A dichotomy. In: 37th International Colloquium on Automata, Languages and Programming, ICALP (to appear 2010)Google Scholar