Goemans and Rothvoss (SODA'14) gave a framework for solving problems in time $enc(P)^{2^{O(N)}}enc(Q)^{O(1)}$ that can be described as finding a point in $\text{int.cone}(P\cap\mathbb{Z}^N)\cap Q$, where $P,Q\subset\mathbb{R}^N$ are (bounded) polyhedra. This framework can be used to solve various scheduling problems, but the encoding length $enc(P)$ usually involves large parameters like the makespan. We describe three tools to improve the framework by Goemans and Rothvoss: Problem-specific preprocessing, LP relaxation techniques and a new bound for the number of vertices of the integer hull. In particular, applied to the classical scheduling problem $P||C_{\max}$, these tools each improve the running time from $(\log(C_{\max}))^{2^{O(d)}} enc(I)^{O(1)}$ to the possibly much better $(\log(p_{\max}))^{2^{O(d)}}enc(I)^{O(1)}$. Here, $p_{\max}$ is the largest processing time, $d$ is the number of different processing times, $C_{\max}$ is the makespan and $enc(I)$ is the encoding length of the instance. This running time is FPT w.r.t. parameter $d$ if $p_{\max}$ is given in unary. We obtain similar results for various other problems. Moreover, we show how a balancing result by Govzmann et al. can be used to speed up an additive approximation scheme by Buchem et al. (ICALP'21) in the high-multiplicity setting. On the complexity side, we use reductions from the literature to provide new parameterized lower bounds for $P||C_{\max}$ and to show that the improved running time of the additive approximation algorithm is probably optimal. Finally, we show that the big open question asked by Mnich and van Bevern (Comput. Oper. Res. '18) whether $P||C_{\max}$ is FPT w.r.t. the number of job types $d$ has the same answer as the question whether $Q||C_{\max}$ is FPT w.r.t. the number of job and machine types $d+\tau$ (all in high-multiplicity encoding). The same holds for objective $C_{\min}$.

翻译：Goemans和Rothvoss（SODA'14）提出了一个框架，用于在时间$enc(P)^{2^{O(N)}}enc(Q)^{O(1)}$内解决可描述为在$\text{int.cone}(P\cap\mathbb{Z}^N)\cap Q$中寻找点的问题，其中$P,Q\subset\mathbb{R}^N$是（有界）多面体。该框架可用于解决多种调度问题，但编码长度$enc(P)$通常涉及大型参数（如制造跨度）。我们描述了三种改进Goemans和Rothvoss框架的工具：问题特定预处理、LP松弛技术以及整数凸包顶点数量的新界限。特别地，应用于经典调度问题$P||C_{\max}$时，这些工具将运行时间从$(\log(C_{\max}))^{2^{O(d)}} enc(I)^{O(1)}$改进为可能更好的$(\log(p_{\max}))^{2^{O(d)}}enc(I)^{O(1)}$。其中$p_{\max}$为最大处理时间，$d$为不同处理时间的数量，$C_{\max}$为制造跨度，$enc(I)$为实例编码长度。若$p_{\max}$以一元形式给出，则此运行时间关于参数$d$为FPT。我们针对其他多种问题获得了类似结果。此外，我们展示了如何利用Govzmann等人的平衡结果加速Buchem等人（ICALP'21）在高多样性设置下的加性近似方案。在复杂性方面，我们利用文献中的归约方法为$P||C_{\max}$提供新的参数化下界，并证明加性近似算法改进后的运行时间可能是最优的。最后，我们证明Mnich和van Bevern（Comput. Oper. Res. '18）提出的核心开放问题——$P||C_{\max}$是否关于作业类型数量$d$为FPT——与$Q||C_{\max}$是否关于作业与机器类型数量$d+\tau$为FPT（均采用高多样性编码）具有相同答案。该结论对目标$C_{\min}$同样成立。