Chapter

Algorithms and Complexity

Volume 6078 of the series Lecture Notes in Computer Science pp 311-322

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

(Extended Abstract)
  • Daniel Binkele-RaibleAffiliated withLancaster UniversityFB 4—Abteilung Informatik, Universität Trier
  • , Ljiljana BrankovicAffiliated withLancaster UniversityThe University of Newcastle
  • , Henning FernauAffiliated withLancaster UniversityFB 4—Abteilung Informatik, Universität Trier
  • , Joachim KneisAffiliated withCarnegie Mellon UniversityDepartment of Computer Science, RWTH Aachen University
  • , Dieter KratschAffiliated withCarnegie Mellon UniversityLaboratoire d’Informatique Théorique et Appliquée, Université Paul Verlaine - Metz
  • , Alexander LangerAffiliated withCarnegie Mellon UniversityDepartment of Computer Science, RWTH Aachen University
  • , Mathieu LiedloffAffiliated withCarnegie Mellon UniversityLaboratoire d’Informatique Fondamentale d’Orléans, Université d’Orléans
  • , Peter RossmanithAffiliated withCarnegie Mellon UniversityDepartment of Computer Science, RWTH Aachen University

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Abstract

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 this open problem by devising parameterized algorithms for the duals of the natural parameterizations of the problems with running times faster than \(\mathcal{O}^*(4^{k})\). For example, we present an algorithm running in time \(\mathcal{O}^*(3.069^{k}))\) 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.