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# Approximating Source Location and Star Survivable Network Problems

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 , 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 , 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 .

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