Finding $k$-Secluded Trees Faster

Abstract

We revisit the k-Secluded Tree problem. Given a vertexweighted 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 O(k log k) • n 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 ′ ⊇ 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.

Finding k-Secluded Trees Faster

“…We prove correctness of the algorithm up to Step 3. The correctness of Step 3 is proven in the full version [5].…”

“…Step 3 ( ). In the full version [5] we show using Properties 1 to 3 that if the algorithm reaches Step 3, then its output is correct. For this we use Lemma 4 to argue that the k-secluded supertrees of F in G can be partitioned into three sets T 1 , T 2 , T 3 .…”