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Approximating Source Location and Star Survivable Network Problems

  • Guy Kortsarz
  • Zeev NutovEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9224)

Abstract

In Source Location (SL) problems the goal is to select a minimum cost source set \(S \subseteq V\) such that the connectivity (or flow) \(\psi (S,v)\) from S to any node v is at least the demand \(d_v\) of v. In many SL problems \(\psi (S,v)=d_v\) if \(v \in S\), so the demand of nodes selected to S is completely satisfied. In a variant suggested recently by Fukunaga [7], every node v selected to S gets a “bonus” \(p_v \le d_v\), and \(\psi (S,v)=p_v+\kappa (S \setminus \{v\},v)\) if \(v \in S\) and \(\psi (S,v)=\kappa (S,v)\) otherwise, where \(\kappa (S,v)\) is the maximum number of internally disjoint (Sv)-paths. While the approximability of many SL problems was seemingly settled to \(\varTheta (\ln d(V))\) in [20], for his variant on undirected graphs Fukunaga achieved ratio \(O(k \ln k)\), where \(k=\max _{v \in V}d_v\) is the maximum demand. We improve this by achieving ratio \(\min \{p^* \ln k,k\} \cdot O(\ln k)\) for a more general version with node capacities, where \(p^*=\max _{v \in V} p_v\) is the maximum bonus. In particular, for the most natural case \(p^*=1\) we improve the ratio from \(O(k \ln k)\) to \(O(\ln ^2k)\). To derive these results, we consider a particular case of the Survivable Network (SN) problem when all edges of positive cost form a star. We obtain ratio \(O(\min \{\ln n,\ln ^2 k\})\) for this variant, improving over the best ratio known for the general case \(O(k^3 \ln n)\) of Chuzhoy and Khanna [3].

In addition, we show that directed SL with unit costs is \(\varOmega (\log n)\)-hard to approximate even for 0, 1 demands, while SL with uniform demands can be solved in polynomial time. Finally, we obtain a logarithmic ratio for a generalization of SL where we also have edge-costs and flow-cost bounds \(\{b_v:v \in V\}\), and require that the minimum cost of a flow of value \(d_v\) from S to every node v is at most \(b_v\).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Rutgers UniversityCamdenUSA
  2. 2.The Open University of IsraelRaananaIsrael

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