Irredundant Set Faster Than O(2ⁿ)

Abstract

In this paper we provide algorithms faster than O(2ⁿ) for two variants of the Irredundant Set problem. More precisely, we give:

• a branch-and-reduce algorithm solving Largest Irredundant Set in $\mathcal{O}(1.9657^{n})$ time and polynomial space; the time complexity can be reduced using memoization to $\mathcal{O}(1.8475^{n})$ at the cost of using exponential space,

• and a simple iterative-DFS algorithm for Smallest Inclusion-Maximal Irredundant Set that solves it in $\mathcal{O}(1.999956^{n})$ time and polynomial space.

Inside the second algorithm time complexity analysis we use a structural approach which allows us to break the O(2ⁿ) barrier. We find this structural approach more interesting than the algorithm itself. Despite the fact that the discussed problems are quite similar to the Dominating Set problem solving them faster than the obvious O(2ⁿ) solution seemed harder; that is why they were posted as an open problems at the Dagstuhl seminar in 2008.

This work was partially supported by the Polish Ministry of Science grants N206 491038 and N206 491238.