AI 中文总结
研究单机调度中最小化\(k\)个最大延迟值总和的问题，证明\(k\)为输入时决策版本弱NP完全，给出固定\(k\)的\(O(k^2 n^{k + 2})\)算法，解决相关猜想并获\(O(n^{k + 2})\)算法，还给出多种算法及结构结果。
AI 中文摘要
我们研究单机调度问题，其中每个作业\(j\)有非负处理时间\(p_j\geq0\)和截止日期\(d_j\in\mathbb{R}\)。对于非空闲调度\(S\)，设\(C_j(S)\)为完成时间，\(L_j(S)=C_j(S)-d_j\)为（可能为负的）延迟。目标是最小化\(k\)个最大延迟值的总和，介于最大延迟（\(k = 1\)）和总延迟（\(k = n\)）之间。我们证明当\(k\)是输入的一部分时，决策版本是弱NP完全的。对于固定的\(k\)，我们给出一个\(O(k^2 n^{k + 2})\)算法。结果，我们解决了Woeginger关于前\(k\)个延迟问题的一个猜想，并为每个固定的\(k\)获得一个\(O(n^{k + 2})\)算法。我们的主要结构结果表明存在一个最优调度，它允许块岛分解。在合适的前\(k\)集之外，作业按截止日期形成截止日期块。在每个截止日期类中，前\(k\)个作业按字典序最短处理时间（SPT）顺序形成一个后缀。这种结构还产生了一个由\(D + k\)参数化的FPT算法，其中\(D\)是不同截止日期的数量。独立地，前\(k\)目标的标准对偶表示将问题简化为一族具有均匀移动截止日期的总延迟实例。对于整数数据，这给出了一个伪多项式算法和一个完全多项式加法近似方案，误差至多为\(\varepsilon M\)，其中\(M=\max\{1,\max_j p_j,\max_j |d_j|\}\)。相同的方法还给出了针对固定\(P\)和固定\(D\)的XP算法，其中\(P\)是不同处理时间的数量。
英文摘要
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.
