The k-Sum Lateness Problem on a Single Machine

2026-07-16

AI Summary

This study addresses the problem of minimizing the sum of the $k$ largest lateness values ($1 \leq k \leq n$) in single-machine non-preemptive scheduling, a measure interpolating between maximum lateness and total lateness. By introducing a novel “block-island” decomposition structure, the work reveals the problem’s optimal substructure and establishes a duality transformation to an equivalent total tardiness problem with uniform release dates. Leveraging this insight, the paper resolves Woeginger’s conjecture on top-$k$ lateness: for fixed $k$, it presents an exact $O(n^{k+2})$ algorithm; for integer inputs, it designs a pseudo-polynomial-time algorithm and an FPTAS achieving error at most $\varepsilon M$; and it yields XP or FPT algorithms when parameterized by $D + k$, $P$, or $D$. The approach integrates techniques from combinatorial optimization, dynamic programming, and parameterized algorithm design.

Abstract

We study a single-machine scheduling problem in which each job $j$ has a nonnegative processing time $p_j\ge 0$ and a due date $d_j\in\mathbb{R}$. For a non-idling schedule $S$, let $C_j(S)$ be the completion time and let $L_j(S)=C_j(S)-d_j$ be the (possibly negative) lateness. The objective is to minimize the sum of the $k$ largest lateness values, interpolating between maximum lateness ($k=1$) and total lateness ($k=n$). We prove that the decision version is weakly NP-complete when $k$ is part of the input. For fixed $k$, we give an $O(k^2 n^{k+2})$ algorithm. As a consequence, we resolve a conjecture of Woeginger on the top-$k$ tardiness problem and obtain an $O(n^{k+2})$ algorithm for every fixed $k$. Our main structural result shows that there exists an optimal schedule that admits a block-island decomposition. Outside a suitable top-$k$ set, jobs form due-date blocks ordered by due date. Within each due-date class, the top-$k$ jobs form a suffix in lexicographic shortest-processing-time (SPT) order. This structure also yields an FPT algorithm parameterized by $D+k$, where $D$ is the number of distinct due dates. Independently, a standard dual representation of the top-$k$ objective reduces the problem to a family of total-tardiness instances with uniformly shifted due dates. For integral data, this gives a pseudopolynomial algorithm and a fully polynomial additive approximation scheme with error at most $\varepsilon M$, where $M=\max\{1,\max_j p_j,\max_j |d_j|\}$. The same route also gives XP algorithms for fixed $P$ and fixed $D$, where $P$ is the number of distinct processing times.

Problem

• k-Sum Lateness
• Single Machine Scheduling
• Lateness Minimization
• Top-k Objective
• NP-completeness

Innovation

• k-sum lateness
• block-island decomposition
• fixed-parameter tractability
• pseudopolynomial algorithm
• scheduling approximation