Approximating Source Location and Star Survivable Network Problems

In Source Location (SL) problems the goal is to select a mini-mum 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 $v \in S$, namely, the demand of nodes selected to $S$ is completely satisfied. In a node-connectivity variant suggested recently by Fukunaga, every node $v$ gets a “bonus” $p_v \leq d_v$ if it is selected to $S$. Fukunaga showed that for undirected graphs one can achieve ratio $O(k \ln k)$ for his variant, where $k=\max_{v \in V}d_v$ is the maximum demand. We improve this by achieving ratio $\min{p^\lnk,k}\cdot O(\ln (k/q^))$ for a more general version with node capacities, where $p^=\max_{v \in V} p_v$ is the maximum bonus and $q^=\min_{v \in V} q_v$ is the minimum capacity. In particular, for the most natural case $p^*=1$ considered by Fukunaga, we improve the ratio from $O(k \ln k)$ to $O(\ln^2k)$. We also get ratio $O(k)$ for the edge-connectivity version, for which no ratio that depends on $k$ only was known before. 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 give 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.

💡 Research Summary

The paper tackles two fundamental network design problems—Source Location (SL) and Survivable Network (SN)—and provides improved approximation algorithms for specific variants that incorporate node bonuses, capacities, and a star-shaped cost structure. In the classic SL problem, one must choose a minimum‑cost set of source vertices S such that for every vertex v the connectivity (or flow) from S to v, denoted ψ(S,v), meets a demand d_v. The standard model assumes ψ(S,v)=d_v automatically when v∈S. Fukunaga introduced a more nuanced version where each selected vertex receives a “bonus” p_v ≤ d_v, which reduces the amount of external connectivity it needs. For undirected graphs, Fukunaga proved an O(k log k) approximation, where k = max_v d_v.

The authors extend this framework in two directions. First, they allow each vertex v to have a capacity q_v, meaning that v can simultaneously support up to q_v edge‑disjoint paths. Second, they focus on a special case of the Survivable Network problem in which all positive‑cost edges form a single star: a central hub connected to a set of leaves, each leaf possibly having its own demand and capacity. By exploiting this star structure, they design algorithms that achieve dramatically better ratios than the best known for the general SN problem (which is O(k³ log n) by Chuzhoy and Khanna).

The main technical contributions are as follows:

1. Node‑connectivity SL with bonuses and capacities.
   The authors obtain an approximation factor of
   \