Approximating Fault-Tolerant Domination in General Graphs

Approximating Fault-Tolerant Domination in General Graphs

Abstract

In this paper we study the NP-complete problem of finding small k-dominating sets in general graphs, which allow k −1 nodes to fail and still dominate the graph. The classic minimum dominating set problem is a special case with k = 1. We show that the approach of having at least k dominating nodes in the neighborhood of every node is not optimal. For each α > 1 it can give solutions k α times larger than a minimum k-dominating set. We also study lower bounds on possible approximation ratios. We show that it is NP-hard to approximate the minimum k-dominating set problem with a factor better than (0.2267/k) ln(n/k). Furthermore, a result for special finite sums allows us to use a greedy approach for k-domination with an approximation ratio of ln(∆ + k) + 1 < ln(∆) + 1.7, with ∆ being the maximum node-degree. We also achieve an approximation ratio of ln(n) + 1.7 for h-step k-domination, where nodes do not need to be direct neighbors of dominating nodes, but can be h steps away.

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Authors
  • Foerster, Klaus-Tycho
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Supplemental Material
Shortfacts
Category
Paper in Conference Proceedings or in Workshop Proceedings (Paper)
Event Title
10th Meeting on Analytic Algorithmics and Combinatorics (ANALCO 2013)
Divisions
Communication Technologies
Subjects
Theoretische Informatik
Event Location
New Orleans, USA
Event Type
Workshop
Event Dates
Jan 6 2013
Date
January 2013
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