First-fit coloring on interval graphs has performance ratio at least 5

TL;DR

This paper investigates the performance ratio of the First-fit online graph coloring algorithm on interval graphs, establishing a new lower bound of at least 5, which advances understanding of its worst-case behavior.

Contribution

The paper proves that the performance ratio of First-fit on interval graphs is at least 5, improving the known lower bound and contributing to the theoretical analysis of online coloring algorithms.

Findings

Established a new lower bound of 5 for the performance ratio.

Improved the understanding of First-fit algorithm's worst-case performance.

Contributed to the theoretical bounds in online graph coloring.

Abstract

First-fit is the online graph coloring algorithm that considers vertices one at a time in some order and assigns each vertex the least positive integer not used already on a neighbor. The maximum number of colors used by first-fit on graph G over all vertex orders is denoted \chi_{FF}(G). The exact value of R := \sup_G [\chi_{FF}(G) / \omega(G)] over interval graphs G is unknown. Pemmaraju, Raman, and Varadarajan (2004) proved R <= 10, and this can be improved to 8. Witsenhausen (1976) and Chrobak and \'Slusarek (1988) showed R >= 4, and \'Slusarek (1993) improved this to 4.45. We prove R >= 5.