Highly Connected Steiner Subgraph - Parameterized Algorithms and Applications to Hitting Set Problems

2023-11-05

arXiv.org

AI Summary

This paper studies the *p-edge-connected Steiner subgraph extension problem*: given a graph $G$, a vertex subset $X$, and integers $k,p$, decide whether there exists a superset $S supseteq X$ with $|S| leq k$ such that the induced subgraph $G[S]$ is $p$-edge-connected. Methodologically, we integrate graph degeneracy analysis, structural exploration of edge connectivity, and dynamic programming with pruning. Our main contributions are threefold: (i) We establish the first systematic parameterized complexity framework for this problem; (ii) We prove it is $W[1]$-hard on general graphs parameterized by $(k,p)$, yet design an $O^*(f(k,p))$-time FPT algorithm for bounded-degeneracy graphs; (iii) Leveraging our structural insights in reverse, we obtain the first single-exponential-time FPT algorithms for several vertex-deletion problems—including $p$-connected feedback vertex set—thereby breaking prior double-exponential time barriers.

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.

Problem

• Extending vertex sets to p-edge-connected subgraphs
• Parameterized algorithms for Steiner subgraph extension
• Applications to vertex deletion with connectivity constraints

Innovation

• Parameterized algorithm for Steiner Subgraph Extension
• Fixed-parameter tractable by k and p parameters
• Applies to vertex deletion problems with connectivity constraints