Fast Algorithms for max independent set

We first propose a 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. Here, 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 successively solve 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 5 and 6 but also for general graphs. The computation bounds obtained for max independent set are O∗(1.1571n), O∗(1.1895n) and O∗(1.2050n), for graphs of maximum (or more generally average) degree 4, 5 and 6 respectively, and O∗(1.2114n) for general graphs. These results improve upon the best known results for these cases for polynomial space algorithms.

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