Heuristics for the data arrangement problem on regular trees

TL;DR

This paper introduces heuristics and a lower bound for the data arrangement problem on regular trees, aiming to improve solution quality for an NP-hard problem by using construction, local search, and performance analysis.

Contribution

It proposes new heuristics and a lower bound for DAPT, along with an analysis framework based on solution quality and optimality gap evaluation.

Findings

Heuristics produce solutions close to the lower bound.

Performance analysis includes solution quality comparison.

Special class instances show the effectiveness of heuristics.

Abstract

The data arrangement problem on regular trees (DAPT) consists in assigning the vertices of a given graph G to the leaves of a d-regular tree T such that the sum of the pairwise distances of all pairs of leaves in T which correspond to edges of G is minimised. Luczak and Noble [6] have shown that this problem is NP-hard for every fixed d larger than or equal to 2. In this paper we propose construction and local search heuristics for the DAPT and introduce a lower bound for this problem. The analysis of the performance of the heuristics is based on two considerations: a) the quality of the solutions produced by the heuristics as compared to the respective lower bounds b) for a special class of instances with known optimal solution we evaluate the gap between the optimal value of the objective function and the objective function value attained by the heuristic solution, respectively.