A Parameterized Route to Exact Puzzles: Breaking the 2n -Barrier for Irredundance

. 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(2ⁿ) enumeration barrier. We solve this open problem by devising parameterized algorithms for the duals of the natural parameterizations 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 shown 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. Furthermore, these seem to be the first examples of a parameterized approach leading to a solution to a problem in exponential time algorithmics where the natural interpretation as exact exponential-time algorithms fails.