We consider the Generalized Makespan Problem (GMP) on unrelated machines, where we are given $n$ jobs and $m$ machines and each job $j$ has arbitrary processing time $p_{ij}$ on machine $i$. Additionally, there is a general symmetric monotone norm $\psi_i$ for each machine $i$, that determines the load on machine $i$ as a function of the sizes of jobs assigned to it. The goal is to assign the jobs to minimize the maximum machine load. Recently, Deng, Li, and Rabani (SODA'22) gave a $3$ approximation for GMP when the $\psi_i$ are top-$k$ norms, and they ask the question whether an $O(1)$ approximation exists for general norms $\psi$? We answer this negatively and show that, under natural complexity assumptions, there is some fixed constant $\delta>0$, such that GMP is $\Omega(\log^{\delta} n)$ hard to approximate. We also give an $\Omega(\log^{1/2} n)$ integrality gap for the natural configuration LP.

翻译：我们考虑无关机器上的广义完工时间问题(GMP)：给定$n$个作业和$m$台机器，每个作业$j$在机器$i$上的处理时间$p_{ij}$是任意的。此外，每台机器$i$有一个一般的对称单调范数$\psi_i$，该范数将分配给该机器的作业规模函数定义为机器负载。目标是将作业分配给机器，以最小化最大机器负载。最近，Deng、Li和Rabani (SODA'22) 针对$\psi_i$为top-$k$范数的情形给出了GMP的$3$近似算法，并提出了一个开放性问题：对于一般范数$\psi$，是否存在$O(1)$近似算法？我们对这个问题给出了否定答案，并证明在自然复杂性假设下，存在某个固定常数$\delta>0$，使得GMP的近似难度为$\Omega(\log^{\delta} n)$。我们还证明了自然配置线性规划松弛的$\Omega(\log^{1/2} n)$整数间隙。