Algorithmic expedients for the S-labeling problem

TL;DR

This paper introduces new algorithmic methods, including an exact MIP framework, for solving the NP-hard S-labeling problem in graphs, providing polynomial solutions for specific graph classes and evaluating their effectiveness computationally.

Contribution

It presents the first polynomial-time algorithm and closed-form solution for the S-labeling problem on n-ary trees, along with enhanced MIP formulations and heuristics.

Findings

MIP formulation has no integrality gap for paths, cycles, and perfect n-ary trees.

First polynomial-time algorithm for S-labeling on n-ary trees.

Computational study shows effectiveness of proposed methods on general graphs.

Abstract

Graph labeling problems have been widely studied in the last decades and have a vast area of application. In this work, we study the recently introduced S-labeling problem, in which the nodes get labeled using labels from 1 to |V | and for each edge the contribution to the objective function, called S-labeling number of the graph, is the minimum label of its end-nodes. The goal is to find a labeling with minimum value. The problem is NP-hard for planar subcubic graphs, although for many other graph classes the complexity status is still unknown. In this paper, we present different algorithmic approaches for tackling this problem: We develop an exact solution framework based on Mixed-Integer Programming (MIP) which is enhanced with valid inequalities, starting and primal heuristics and specialized branching rules. We show that our MIP formulation has no integrality gap for paths, cycles and perfect n-ary trees, and, to the best of our knowledge, we give the first polynomial-time algorithm for the problem on n-ary trees as well as a closed formula for the S-labeling number for such trees. Moreover, we also present a Lagrangian heuristic and a constraint programming approach. A computational study is carried out in order to (i) investigate if there may be other special graph classes, where our MIP formulation has no integrality gap, and (ii) assess the effectiveness of the proposed solution approaches for solving the problem on a dataset consisting of general graphs.