Parameterized algorithms and data reduction for the short secluded $s$-$t$-path problem

TL;DR

This paper investigates the parameterized complexity of the Short Secluded Path problem, providing kernelization results and algorithms based on structural graph parameters such as treewidth and vertex cover.

Contribution

It offers a complete classification of polynomial kernel existence for the problem relative to key graph parameters and introduces a new algorithm with subexponential runtime for certain graph classes.

Findings

No polynomial kernels for certain parameters unless complexity collapses.

A $2^{O(w)} imes ext{poly}( ext{n}, ext{ell})$-time algorithm for graphs of bounded treewidth.

Subexponential algorithms for specific graph classes.

Abstract

Given a graph $G=(V,E)$, two vertices $s,t\in V$, and two integers $k,\ell$, the Short Secluded Path problem is to find a simple $s$-$t$-path with at most $k$ vertices and $\ell$ neighbors. We study the parameterized complexity of the problem with respect to four structural graph parameters: the vertex cover number, treewidth, feedback vertex number, and feedback edge number. In particular, we completely settle the question of the existence of problem kernels with size polynomial in these parameters and their combinations with $k$ and $\ell$. We also obtain a $2^{O(w)}\cdot \ell^2\cdot n$-time algorithm for graphs of treewidth $w$, which yields subexponential-time algorithms in several graph classes.