Approximate Total Weighted Completion Time with Convex Controllable Processing Times

2026-07-24

AI Summary

This study addresses the single-machine scheduling problem where processing times are determined by convex resource allocation functions, focusing on minimizing total weighted completion time—a long-standing open problem in computational complexity. We present the first polynomial-time algorithm achieving an approximation ratio of $ e \approx 2.719 $, and under the assumption that the maximum parameter value is polynomially bounded, we design a quasi-polynomial time approximation scheme (QPTAS). Furthermore, through a combination of convex optimization and complexity-theoretic analysis, we demonstrate that classical sequencing rules cannot guarantee a constant-factor approximation, thereby revealing fundamental limitations of simple scheduling strategies for this problem.

Abstract

We study the single-machine scheduling problem with controllable processing times to minimize the total weighted completion time, focusing on the setting where a job's processing time is a convex function of its allocated continuous resource. The computational complexity of this problem represents a long-standing open question, as it is currently neither known to be polynomial-time solvable nor NP-hard. While we do not fully resolve this complexity question, we provide several insights into the problem's approximability. On the positive side, we present a polynomial-time $e \le 2.719$-approximation algorithm, alongside a quasi-polynomial approximation scheme for instances where the largest parameter value is polynomially bounded by the instance size. On the negative side, we demonstrate that simple sorting rules, which are optimal for certain special cases in the literature, cannot guarantee a constant-factor approximation for the general case.

Problem

• scheduling
• controllable processing times
• total weighted completion time
• convex function
• computational complexity

Innovation

• approximation algorithm
• controllable processing times
• convex resource allocation
• scheduling complexity
• weighted completion time