Approximating Rooted Connectivity Augmentation Problems

• Authors: Zeev Nutov
• Venue: Algorithmica
• Year: 2006

A graph is called {\em $\el$-connected from $U$ to $r$} if there are $\el$ internally disjoint paths from every node $u \in U$ to $r$. The {\em Rooted Subset Connectivity Augmentation Problem} ({\em RSCAP}) is as follows: given a graph $G=(V+r,E)$, a node subset $U \subseteq V$, and an integer $k$, find a smallest set $F$ of new edges such that $G+F$ is $k$-connected from $U$ to $r$. In this paper we consider mainly a restricted version of RSCAP in which the input graph $G$ is already $(k-1)$-connected from $U$ to $r$. For this version we give an $O(\ln\! |U|)$-approximation algorithm, and show that the problem cannot achieve a better approximation guarantee than the Set Cover Problem (SCP) on $|U|$ elements and with $|V|-|U|$ sets. For the general version of RSCAP we give an $O(\ln k \ln\!|U|)$-approximation algorithm. For $U=V$ we get the {\em Rooted Connectivity Augmentation Problem} ({\em RCAP}). For directed graphs RCAP is polynomially solvable, but for undirected graphs its complexity status is not known: no polynomial algorithm is known, and it is also not known to be NP-hard. For undirected graphs with the input graph $G$ being $(k-1)$-connected from $V$ to $r$, we give an algorithm that computes a solution of size at most ${\it opt}+\min\{opt,k\}/2$, where {\it opt} denotes the optimal solution size.