A Lower Bound on the Competitive Ratio of the Permutation Algorithm for Online Facility Assignment on a Line
Tsubasa Harada
TL;DR
This paper studies online facility assignment on a line (OFAL) with $k$ servers and equal capacities, focusing on the permutation algorithm. It proves that the competitive ratio of the permutation algorithm for OFAL$_{eq}$ with evenly placed servers is at least $k+1$, contradicting the previously claimed $k$-competitiveness and addressing a known discrepancy for small $k$. The authors establish a reduction showing ${\cal R}_{k,\ell}(\alg)\ge {\cal R}_{k,1}(\alg)$ and construct a worst-case request sequence to show ${\rm perm}$ achieves at least $(k+1-\epsilon)$ of ${\rm opt}$ for any $\epsilon>0$, in both parity cases of $k$. The work suggests the exact bound is $k+1$ and highlights that capacities may affect the ratio, motivating further investigation into tightness and capacity dependence.
Abstract
In the online facility assignment on a line (OFAL) with a set $S$ of $k$ servers and a capacity $c:S\to\mathbb{N}$, each server $s\in S$ with a capacity $c(s)$ is placed on a line and a request arrives on a line one-by-one. The task of an online algorithm is to irrevocably assign a current request to one of the servers with vacancies before the next request arrives. An algorithm can assign up to $c(s)$ requests to each server $s\in S$. In this paper, we show that the competitive ratio of the permutation algorithm is at least $k+1$ for OFAL where the servers are evenly placed on a line. This disproves the result that the permutation algorithm is $k$-competitive by Ahmed et al..