Competitive Kill-and-Restart and Preemptive Strategies for Non-Clairvoyant Scheduling

TL;DR

This paper analyzes non-clairvoyant scheduling strategies, establishing lower bounds and providing tight competitive ratios for kill-and-restart and preemptive algorithms, with implications for online and parallel machine settings.

Contribution

It introduces tight bounds and competitive ratios for kill-and-restart and preemptive scheduling strategies in non-clairvoyant settings, including new randomized and deterministic analyses.

Findings

Deterministic kill-and-restart has a lower bound ratio of 3.

A tight competitive ratio of approximately 6.197 is achieved for deterministic strategies.

Preemptive WSETF is 2-competitive for online jobs, matching lower bounds.

Abstract

We study kill-and-restart and preemptive strategies for the fundamental scheduling problem of minimizing the sum of weighted completion times on a single machine in the non-clairvoyant setting. First, we show a lower bound of~$3$ for any deterministic non-clairvoyant kill-and-restart strategy. Then, we give for any $b > 1$ a tight analysis for the natural $b$-scaling kill-and-restart strategy as well as for a randomized variant of it. In particular, we show a competitive ratio of $(1+3\sqrt{3})\approx 6.197$ for the deterministic and of $\approx 3.032$ for the randomized strategy, by making use of the largest eigenvalue of a Toeplitz matrix. In addition, we show that the preemptive Weighted Shortest Elapsed Time First (WSETF) rule is $2$-competitive when jobs are released online, matching the lower bound for the unit weight case with trivial release dates for any non-clairvoyant algorithm. Using this result as well as the competitiveness of round-robin for multiple machines, we prove performance guarantees smaller than $10$ for adaptions of the $b$-scaling strategy to online release dates and unweighted jobs on identical parallel machines.