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

02/01/2021

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by   Masato Tanaka, et al.

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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.