The k-Sum Lateness Problem on a Single Machine
Data Structures and Algorithms
Summary
The authors study how to schedule jobs on one machine to minimize the sum of the largest lateness values, which is a middle ground between minimizing just the worst lateness and all lateness. They show the problem is hard when the number of top latenesses to consider (k) varies, but provide an algorithm when k is fixed and confirm a previous conjecture for this case. They also find a way to organize jobs to make scheduling easier and develop additional algorithms based on the number of unique due dates or processing times. Their work includes both exact and approximate methods depending on the problem parameters.
Authors
Ricardo Arancibia-Castillo, José A. Soto
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.