Online coloring of short intervals

Abstract

We study the online graph coloring problem restricted to the intersection graphs of intervals with lengths in [1,σ]. For σ=1 it is the class of unit interval graphs and for σ=∞ the class of all interval graphs. Our focus is on intermediary classes. We present a (1+σ)-competitive algorithm, which beats the state of the art for 1<σ<2, and proves that the problem we study can be strictly easier than online coloring of general interval graphs. On the lower bound side, we prove that no algorithm is better than 5/3-competitive for any σ>1, nor better than 7/4-competitive for any σ>2, and that no algorithm beats the 5/2 asymptotic competitive ratio for all, arbitrarily large, values of σ. That last result shows that the problem we study can be strictly harder than unit interval coloring. Our main technical contribution is a recursive composition of strategies, which seems essential to prove any lower bound higher than 2. Moreover, we prove that the natural algorithm FirstFit is no better than 5/2 for any σ>3, no better than 11/4 for any σ>12, and finally for every ɛ>0 FirstFit is no better than (5−ɛ)-competitive for any σ>σɛ (where σɛ is some finite length depending on ɛ). We also give some upper bounds for FirstFit for σ∈[5,13].