Breaking the 2^n-Barrier for Irredundance: A Parameterized Route to Solving Exact Puzzles [article]

Ljiljana Brankovic, Henning Fernau, Joachim Kneis, Dieter Kratsch Alexander Langer Mathieu Liedloff Daniel Raible Peter Rossmanith
2009 arXiv   pre-print
The lower and the upper irredundance numbers of a graph G, denoted ir(G) and IR(G) respectively, are conceptually linked to domination and independence numbers and have numerous relations to other graph parameters. It is a long-standing open question whether determining these numbers for a graph G on n vertices admits exact algorithms running in time less than the trivial Ω(2^n) enumeration barrier. We solve these open problems by devising parameterized algorithms for the dual of the natural
more » ... ameterizations of the problems with running times faster than O^*(4^k). For example, we present an algorithm running in time O^*(3.069^k) for determining whether IR(G) is at least n-k. Although the corresponding problem has been known to be in FPT by kernelization techniques, this paper offers the first parameterized algorithms with an exponential dependency on the parameter in the running time. Additionally, our work also appears to be the first example of a parameterized approach leading to a solution to a problem in exponential time algorithmics where the natural interpretation as an exact exponential-time algorithm fails.
arXiv:0909.4224v1 fatcat:bvjx4me43zgt5ddhvc7bsaqwcq