Approximating Source Location and Star Survivable Network Problems

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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 S,i¾?v-paths. While the approximability of many SL problems was seemingly settled to $$\varTheta \ln dV$$ in [20], for his variant on undirected graphs Fukunaga achieved ratio $$Ok \ln k$$, 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 $$Ok \ln k$$ 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 $$Ok^3 \ln n$$ of Chuzhoy and Khannai¾?[3]. In addition, we show that directed SL with unit costs is $$\varOmega \log n$$-hard to approximate even for 0,i¾?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 $$b_v$$.