The tight absolute bound of First Fit in the parameterized case

Abstract

The tight asymptotic approximation ratio of the classical bin packing algorithm called First Fit was known for many years, also in the parameterized case, when any item has size of at most $1/d$, where d is integer. But the tight absolute bound was found only recently, and only for $d=1$. Here we give the tight absolute approximation ratio of First Fit for any $d\ge 2$. In fact, we do more. For any value of OPT (the number of bins in an optimum solution) we determine that exactly how large FF (the number of bins created by First Fit) can be in the worst case.

The proof is very simple, since we can apply the lower bound construction of case $d=1$, in a slightly modified form.