Highly Connected Steiner Subgraph -- Parameterized Algorithms and Applications to Hitting Set Problems
Eduard Eiben, Diptapriyo Majumdar, M. S. Ramanujan
TL;DR
The paper addresses Steiner Subgraph Extension, asking for a $p$-edge-connected Steiner subgraph on at most $k$ vertices containing the given terminals, parameterized by $k+p$. It proves a fixed-parameter tractable algorithm on $\eta$-degenerate graphs with runtime $2^{\mathcal{O}(pk+\eta)}n^{\mathcal{O}(1)}$, using a novel combination of out-partition matroids and representative sets within a dynamic-programming framework over a degeneracy ordering, and shows W[1]-hardness without a degeneracy bound. It then leverages this main result to obtain singly exponential-time FPT algorithms for several vertex-deletion hitting-set problems with $p$-edge-connectivity constraints, across various target hereditary graph classes (e.g., $\eta$-degenerate, pathwidth-1, treedepth-bounded), with explicit runtimes and structural conditions. The work also connects to Edge-Connected Survivable Network Design and provides corollaries for treewidth and cutwidth-bounded graphs, highlighting practical implications for hitting-set style problems on graphs and offering several avenues for future research on broader graph classes and related connectivity variants.
Abstract
Given a simple connected undirected graph G = (V, E), a set X \subseteq V(G), and integers k and p, STEINER SUBGRAPH EXTENSION problem asks if there exists a set S \supseteq X with at most k vertices such that G[S] is p-edge-connected. This is a natural generalization of a well-studied problem STEINER TREE (set p=1 and X as the set of all terminals). In this paper, we initiate the study of STEINER SUBGRAPH EXTENSION from the perspective of parameterized complexity and give a fixed-parameter algorithm parameterized by k and p on graphs of bounded degeneracy. In case we remove the assumption of the input graph being bounded degenerate, then the STEINER SUBGRAPH EXTENSION problem becomes W[1]-hard. Besides being an independent advance on the parameterized complexity of network design problems, our result has natural applications. In particular, we use our result to obtain singly exponential-time FPT algorithms for several vertex deletion problem studied in the literature, where the goal is to delete a smallest set of vertices such that (i) the resulting graph belongs to a specific hereditary graph class, and (ii) the deleted set of vertices induces a p-edge-connected subgraph of the input graph.