The data arrangement problem on binary trees

TL;DR

This paper proves the NP-hardness of the data arrangement problem on trees even when guest graphs are trees, and introduces a 1.015-approximation algorithm for a special case involving binary regular trees.

Contribution

It establishes NP-hardness for the problem with tree guest graphs and provides a near-optimal approximation algorithm for binary regular trees.

Findings

NP-hardness proven for tree guest graphs

A 1.015-approximation algorithm for binary regular trees

Closed-form solution and bounds for the approximation algorithm

Abstract

The data arrangement problem on regular trees (DAPT) consists in assigning the vertices of a given graph G, called the guest graph, to the leaves of a d-regular tree T, called the host graph, such that the sum of the pairwise distances of all pairs of leaves in T which correspond to the edges of G is minimised. Luczak and Noble have shown that this problem is NP-hard for every fixed d greater than or equal to 2. In this paper we show that the DAPT remains NP-hard even if the guest graph is a tree, an issue which was posed as an open question in by Luczak and Noble. We deal with a special case of the DAPT where both the guest and the host graph are binary regular trees and provide a 1.015-approximation algorithm for this special case. The solution produced by the algorithm and the corresponding value of the objective function are given in closed form. The analysis of the approximation algorithm involves an auxiliary problem which is interesting on its own, namely the k-balanced partitioning problem (kBPP) for binary regular trees and particular choices of k. We derive a lower bound for the later problem and obtain a lower bound for the original problem by solving hG instances of the k-BPP, where hG is the height of the host graph G.