Strong NP-Hardness and Approximation Algorithm for Weighted Tardiness with Release Dates and Identical Processing Times

2026-09-23

AI Summary

This study addresses the long-standing open problem regarding the computational complexity of minimizing total weighted tardiness in single-machine scheduling, which has remained unresolved since 2010. By integrating computational complexity theory with combinatorial optimization techniques, this work establishes a constructive canonicalization theorem via a reduction from MAX-CUT, thereby proving for the first time that the problem is strongly NP-complete. Furthermore, it proposes a deterministic polynomial-time phase-grid allocation algorithm. This research definitively resolves the complexity question that has persisted for over a decade. The proposed algorithm achieves a tight approximation ratio of 3/2 − 1/(2N) within O(N^5) time complexity, providing a critical theoretical and algorithmic foundation for related research in scheduling optimization.

Abstract

We study nonpreemptive scheduling on a single machine with release dates, due dates, positive job weights, and a common processing time. The objective is to minimize total weighted tardiness. Although closely related equal-processing-time problems admit polynomial-time algorithms, the complexity of this problem has remained open in the literature since 2010. We prove that its decision version is strongly NP-complete, even when every job can meet its due date if processed immediately upon release. The reduction is from unweighted MAX-CUT and uses a quadratic number of jobs with polynomially bounded numerical data. Its main ingredient is a constructive normalization theorem that converts every sufficiently inexpensive feasible schedule into a binary choice for each graph vertex; after normalization, total weighted tardiness equals a constant minus a scaled cut value. We also give a deterministic polynomial-time phase-grid assignment algorithm for the shifted objective $Φ=F+p\sum_jw_j$, where $F$ is total weighted tardiness. The algorithm enumerates at most $N$ release-date residues modulo $p$, solves one minimum-cost assignment problem for each residue, and returns the best phase-grid schedule. It runs in $O(N^5)$ arithmetic operations and achieves the tight ratio $3/2-1/(2N)$ for this algorithm. Because the added term $p\sum_jw_j$ is independent of how the jobs are scheduled, the shifted and original objectives have exactly the same optimal schedules. However, the approximation guarantee applies to the shifted objective; for the original objective, the analysis provides an additive bound. Thus, the paper both resolves the long-standing complexity question and provides a complementary worst-case guarantee for the phase-grid assignment algorithm.

Problem

• single machine scheduling
• weighted tardiness
• release dates
• strong NP-hardness
• approximation algorithm

Innovation

• Strong NP-Hardness
• Weighted Tardiness
• Approximation Algorithm
• Phase-Grid Assignment
• MAX-CUT Reduction