# Polynomial Time Efficient Construction Heuristics for Vertex Separation Minimization Problem

## TL;DR

This paper introduces three polynomial-time construction heuristics for the Vertex Separation Minimization Problem, demonstrating significant improvements over existing heuristics and competitive results with metaheuristics across various graph instances.

## Contribution

Proposes three new polynomial-time heuristics for VSMP and evaluates their effectiveness against existing methods and metaheuristics on diverse graph datasets.

## Key findings

- H1, H2, and H3 achieve best results on 88.71%, 43.5%, and 37.1% of instances respectively.
- H1 outperforms the best literature heuristic on 39.9% of instances.
- H1's results are comparable to the metaheuristic GVNS, which achieves the best results on 82.9% of instances.

## Abstract

Vertex Separation Minimization Problem (VSMP) consists of finding a layout of a graph G = (V,E) which minimizes the maximum vertex cut or separation of a layout. It is an NP-complete problem in general for which metaheuristic techniques can be applied to find near optimal solution. VSMP has applications in VLSI design, graph drawing and computer language compiler design. VSMP is polynomially solvable for grids, trees, permutation graphs and cographs. Construction heuristics play a very important role in the metaheuristic techniques as they are responsible for generating initial solutions which lead to fast convergence. In this paper, we have proposed three construction heuristics H1, H2 and H3 and performed experiments on Grids, Small graphs, Trees and Harwell Boeing graphs, totaling 248 instances of graphs. Experiments reveal that H1, H2 and H3 are able to achieve best results for 88.71%, 43.5% and 37.1% of the total instances respectively while the best construction heuristic in the literature achieves the best solution for 39.9% of the total instances. We have also compared the results with the state-of-the-art metaheuristic GVNS and observed that the proposed construction heuristics improves the results for some of the input instances. It was found that GVNS obtained best results for 82.9% instances of all input instances and the heuristic H1 obtained best results for 82.3% of all input instances.