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 $k$, 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 Björklund, 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).