New Formulation for Coloring Circle Graphs and its Application to Capacitated Stowage Stack Minimization

TL;DR

This paper introduces a new integer linear programming approach for coloring circle graphs, enabling efficient calculation of fractional chromatic numbers and applying this method to optimize capacitated stowage stack minimization.

Contribution

It presents a novel ILP formulation for circle graph coloring and extends it to a capacitated stowage stack minimization problem, with polynomial-sized LP for fractional chromatic number.

Findings

Linear relaxation finds fractional chromatic number

Polynomial-sized LP formulation developed

Application to capacitated stowage stack minimization

Abstract

A circle graph is a graph in which the adjacency of vertices can be represented as the intersection of chords of a circle. The problem of calculating the chromatic number is known to be NP-complete, even on circle graphs. In this paper, we propose a new integer linear programming formulation for a coloring problem on circle graphs. We also show that the linear relaxation problem of our formulation finds the fractional chromatic number of a given circle graph. As a byproduct, our formulation gives a polynomial-sized linear programming formulation for calculating the fractional chromatic number of a circle graph. We also extend our result to a formulation for a capacitated stowage stack minimization problem.