SWAT 2010: Algorithm Theory - SWAT 2010 pp 62-73

A Bottom-Up Method and Fast Algorithms for max independent set

• Nicolas Bourgeois
• Bruno Escoffier
• Vangelis Th. Paschos
• Johan M. M. van Rooij
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6139)

Abstract

We first propose a new method, called “bottom-up method”, that, informally, “propagates” improvement of the worst-case complexity for “sparse” instances to “denser” ones and we show an easy though non-trivial application of it to the min set cover problem. We then tackle max independent set. Following the bottom-up method we propagate improvements of worst-case complexity from graphs of average degree d to graphs of average degree greater than d. Indeed, using algorithms for max independent set in graphs of average degree 3, we tackle max independent set in graphs of average degree 4, 5 and 6. Then, we combine the bottom-up technique with measure and conquer techniques to get improved running times for graphs of maximum degree 4, 5 and 6 but also for general graphs. The best computation bounds obtained for max independent set are O*(1.1571n), O*(1.1918n) and O*(1.2071n), for graphs of maximum (or more generally average) degree 4, 5 and 6 respectively, and O*(1.2127n) for general graphs. These results improve upon the best known polynomial space results for these cases.

Keywords

Bottom-Up Method Max Independent Set Exact Algorithms

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Authors and Affiliations

• Nicolas Bourgeois
• 1
• Bruno Escoffier
• 1
• Vangelis Th. Paschos
• 1
• Johan M. M. van Rooij
• 2
1. 1.LAMSADECNRS FRE 3234 and Université Paris-DauphineFrance
2. 2.Department of Information and Computing SciencesUniversiteit UtrechtThe Netherlands