Finding $k$-Secluded Trees Faster

TL;DR

This paper introduces a significantly faster fixed-parameter tractable algorithm for the extsc{$k$-Secluded Tree} problem, improving computational efficiency and enabling enumeration and counting of optimal solutions in weighted graphs.

Contribution

The authors develop a new algorithm with improved running time for finding maximum-weight $k$-secluded trees, including structural insights and enumeration capabilities.

Findings

Algorithm runs in $2^{O(k \, \log k)} \cdot n^{O(1)}$ time

Structural result bounds branching depth in the algorithm

Allows enumeration and counting of maximum-weight $k$-secluded trees

Abstract

We revisit the \textsc{$k$-Secluded Tree} problem. Given a vertex-weighted undirected graph $G$, its objective is to find a maximum-weight induced subtree $T$ whose open neighborhood has size at most $k$. We present a fixed-parameter tractable algorithm that solves the problem in time $2^{\mathcal{O}(k \log k)}\cdot n^{\mathcal{O}(1)}$, improving on a double-exponential running time from earlier work by Golovach, Heggernes, Lima, and Montealegre. Starting from a single vertex, our algorithm grows a $k$-secluded tree by branching on vertices in the open neighborhood of the current tree $T$. To bound the branching depth, we prove a structural result that can be used to identify a vertex that belongs to the neighborhood of any $k$-secluded supertree $T' \supseteq T$ once the open neighborhood of $T$ becomes sufficiently large. We extend the algorithm to enumerate compact descriptions of all maximum-weight $k$-secluded trees, which allows us to count the number of maximum-weight $k$-secluded trees containing a specified vertex in the same running time.