Algorithms for 2-connected network design and flexible Steiner trees with a constant number of terminals

TL;DR

This paper introduces randomized algorithms and a PTAS for solving the k-Steiner-2NCS and k-Steiner-2ECS problems, focusing on finding minimum-cost two-node or two-edge connected subgraphs containing a fixed set of terminals.

Contribution

It provides the first polynomial-time randomized algorithms and a PTAS for these network design problems with a constant number of terminals, extending previous cycle-finding techniques.

Findings

Polynomial-time randomized algorithm for unweighted k-Steiner-2NCS.

PTAS for weighted k-Steiner-2NCS.

Similar results achieved for k-Steiner-2ECS problem.

Abstract

The $k$-Steiner-2NCS problem is as follows: Given a constant $k$, and an undirected connected graph $G = (V,E)$, non-negative costs $c$ on $E$, and a partition $(T, V-T)$ of $V$ into a set of terminals, $T$, and a set of non-terminals (or, Steiner nodes), where $|T|=k$, find a minimum-cost two-node connected subgraph that contains the terminals. We present a randomized polynomial-time algorithm for the unweighted problem, and a randomized PTAS for the weighted problem. We obtain similar results for the $k$-Steiner-2ECS problem, where the input is the same, and the algorithmic goal is to find a minimum-cost two-edge connected subgraph that contains the terminals. Our methods build on results by Bj\"orklund, Husfeldt, and Taslaman (ACM-SIAM SODA 2012) that give a randomized polynomial-time algorithm for the unweighted $k$-Steiner-cycle problem; this problem has the same inputs as the unweighted $k$-Steiner-2NCS problem, and the algorithmic goal is to find a minimum-size simple cycle $C$ that contains the terminals ($C$ may contain any number of Steiner nodes).