Minimum Certificate Dispersal with Tree Structures

TL;DR

This paper studies the complexity of the Minimum Certificate Dispersal problem when requests form tree structures, revealing hardness results and polynomial-time solutions depending on the tree's maximum degree and the graph's structure.

Contribution

It proves APX-hardness for tree request sets, analyzes approximation bounds based on tree degree, and identifies polynomial-time solvability when both graph and requests are trees.

Findings

MCD is APX-hard for tree requests, even stars.

Constant degree trees allow polynomial solutions.

G being a tree enables polynomial-time solutions.

Abstract

Given an n-vertex graph G=(V,E) and a set R \subseteq {{x,y} | x,y \in V} of requests, we consider to assign a set of edges to each vertex in G so that for every request {u, v} in R the union of the edge sets assigned to u and v contains a path from u to v. The Minimum Certificate Dispersal Problem (MCD) is defined as one to find an assignment that minimizes the sum of the cardinality of the edge set assigned to each vertex. This problem has been shown to be LOGAPX-complete for the most general setting, and APX-hard and 2-approximable in polynomial time for dense request sets, where R forms a clique. In this paper, we investigate the complexity of MCD with sparse (tree) structures. We first show that MCD is APX-hard when R is a tree, even a star. We then explore the problem from the viewpoint of the maximum degree \Delta of the tree: MCD for tree request set with constant \Delta is solvable in polynomial time, while that with \Delta=\Omega(n) is 2.56-approximable in polynomial time but hard to approximate within 1.01 unless P=NP. As for the structure of G itself, we show that the problem can be solved in polynomial time if G is a tree.