On the Complexity of Secluded Path Problems

TL;DR

This paper explores the computational complexity of finding secluded paths in graphs, introducing new variants and algorithms, and analyzing their tractability under various parameters and graph types.

Contribution

It introduces the extsc{Shortest Secluded Path} problem, analyzes its complexity, and provides new fixed-parameter algorithms and polynomial-time solutions for specific cases.

Findings

The extsc{Short Secluded Path} problem is W[1]-hard when parameterized by path length or cliquewidth.

An XP algorithm parameterized by cliquewidth is developed.

The shortest secluded path variant is polynomial-time solvable on unweighted graphs.

Abstract

This paper investigates the complexity of finding secluded paths in graphs. We focus on the \textsc{Short Secluded Path} problem and a natural new variant we introduce, \textsc{Shortest Secluded Path}. Formally, given an undirected graph $G=(V, E)$, two vertices $s,t\in V$, and two integers $k,l$, the \textsc{Short Secluded Path} problem asks whether there exists an $s$-$t$ path of length at most $k$ with at most $l$ neighbors. This problem is known to be computationally hard: it is W[1]-hard when parameterized by the path length $k$ or by cliquewidth, and para-NP-complete when parameterized by the number $l$ of neighbors. The fixed-parameter tractability is known for $k+l$ or treewidth. In this paper, we expand the parameterized complexity landscape by designing (1) an XP algorithm parameterized by cliquewidth and (2) fixed-parameter algorithms parameterized by neighborhood diversity and twin cover number, respectively. As a byproduct, our results also yield parameterized algorithms for the classic \textsc{$s$-$t$ $k$-Path} problem under the considered parameters. Furthermore, we introduce the \textsc{Shortest Secluded Path} problem, which seeks a shortest $s$-$t$ path with the minimum number of neighbors. In contrast to the hardness of the original problem, we reveal that this variant is solvable in polynomial time on unweighted graphs. We complete this by showing that for edge-weighted graphs, the problem becomes W[1]-hard yet remains in XP when parameterized by the shortest path distance between $s$ and $t$.