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

The k-Steiner-2NCS problem is as follows: Given a constant (positive integer) k, and an undirected connected graph G = (V, E), non-negative costs c on the edges, 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 min-cost two-node connected subgraph that contains the terminals. The k-Steiner-2ECS problem has the same inputs; the algorithmic goal is to find a min-cost two-edge connected subgraph that contains the terminals. We present a randomized polynomial-time algorithm for the unweighted k-Steiner-2NCS problem, and a randomized FPTAS for the weighted k-Steiner-2NCS problem. We obtain similar results for a capacitated generalization of the k-Steiner-2ECS problem. Our methods build on results by Björklund, Husfeldt, and Taslaman (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 min-cost simple cycle C that contains the terminals (C may contain any number of Steiner nodes).