Faster Minimization of Total Weighted Completion Time on Parallel Machines
Danny Hermelin, Tomohiro Koana, Dvir Shabtay
TL;DR
This work addresses the classic scheduling problem $Pm||\sum w_j C_j$ on $m$ identical machines, which is weakly NP-hard and previously approached via Lawler–Moore's pseudo-polynomial DP. The authors prove a structural theorem bounding load deviations in optimal WSPT schedules and develop a dynamic-programming framework that uses load-deviation vectors and Minkowski sums to achieve a pseudo-polynomial time algorithm with runtime $\tilde{O}(P^{m-1} + w_{\max}^{m+1} \cdot p_{\max}^{4m+1})$, improving over Lawler–Moore in regimes where $w_{\max}$ and $p_{\max}$ are small. A refined two-machine result yields a tighter bound of $\tilde{O}(P + w_{\max}^{2} p_{\max}^{5})$, enhancing efficiency for $P2||\sum w_j C_j$. The approach hinges on a structure-based reduction to a low-radius search over load deviations $\Delta_{i,\ell}$ (with radius $c=O(\sqrt{w_{\max}}\, p_{\max}^{2})$), enabling an exact DP and shedding light on the optimal WSPT configurations; these results advance exact algorithms for this fundamental scheduling problem and open avenues for further parameterized improvements.
Abstract
We study the classical problem of minimizing the total weighted completion time on a fixed set of $m$ identical machines working in parallel, the $Pm||\sum w_jC_j$ problem in the standard three field notation for scheduling problems. This problem is well known to be NP-hard, but only in the ordinary sense, and appears as one of the fundamental problems in any scheduling textbook. In particular, the problem served as a proof of concept for applying pseudo-polynomial time algorithms and approximation schemes to scheduling problems. The fastest known pseudo-polynomial time algorithm for $Pm||\sum w_jC_j$ is the famous Lawler and Moore algorithm from the late 1960's which runs in $\tilde{O}(P^{m-1}n)$ time, where $P$ is the total processing time of all jobs in the input. After more than 50 years, we are the first to present an algorithm, alternative to that of Lawler and Moore, which is faster for certain range of the problem parameters (e.g., when their values are all $O(1)$).