Let there be a set M of m parallel machines and a set J of n jobs, where each job j takes $p_{i,j}$ time units on machine $M_i$. In makespan minimization the goal is to schedule each job non-preemptively on a machine such that the length of the schedule, the makespan, is minimum. We investigate a generalization of makespan minimization on unrelated parallel machines ($R||C_{max}$) where J is partitioned into b bags $B=(B_1,\dots ,B_b)$, and no two jobs belonging to the same bag can be scheduled on the same machine. First we present a simple b-approximation algorithm for $R||C_{max}$ with bags ($R|bag|C_{max}$). We also give a polynomial-time approximation scheme (PTAS) for $R|bag|C_{max}$ with machine types where both the number of machine types and bags are constant; two machines $M_i$ and $M_{i^{\prime }}$ have the same machine type if $p_{i,j}=p_{i^{\prime },j}$ for all $j\in J$. This result infers the existence of a PTAS for uniform parallel machines ($Q|bag|C_{max}$) when the number of machine speeds and number of bags are both constant. Then, we present a $b/2$-approximation algorithm for the graph balancing problem with $b\ge 2$ bags; the approximation ratio is tight for $b=3$, unless $\mathsf{P}=\mathsf{NP}$, and this algorithm solves the graph balancing problem with $b=2$ bags in polynomial time. In addition, we present a polynomial-time algorithm for the restricted assignment problem on uniform parallel machines with bags when all the jobs have unit length. To complement our algorithmic results, we show that when the jobs have lengths 1 or 2 it is $\mathsf{NP}$-hard to approximate the optimum makespan within a factor smaller than 3/2 for both the restricted assignment and graph balancing problems with $b=2$ bags and $b=3$ bags, respectively. We also prove that $Q|bag|C_{max}$ with $b=2$ bags is strongly $\mathsf{NP}$-hard.

要有一组中号的米平行机和一组Ĵ的Ñ作业，其中，每个作业Ĵ取$p_{\mathrm{一世}，Ĵ}$ 机器上的时间单位 ${\mathrm{中号}}_{\mathrm{一世}}$。在使工期最小化中，目标是在机器上非抢先地调度每个作业，以使工期长度（工期）最小。我们调查了无关并行计算机上的makepan最小化的一般化（$\mathrm{[R}||C_{米\mathrm{一种}X}$）其中J分为b袋$乙=（{乙}_{\mathrm{1个}}，\dots ，{乙}_b）$，并且不能在同一台计算机上安排属于同一包的两个作业。首先，我们提出一种简单的b-近似算法$\mathrm{[R}||C_{米\mathrm{一种}X}$ 带袋（$\mathrm{[R}|b\mathrm{一种}G|C_{米\mathrm{一种}X}$）。我们还给出了多项式时间近似方案（PTAS）$\mathrm{[R}|b\mathrm{一种}G|C_{米\mathrm{一种}X}$机器类型和袋子数量均恒定的机器类型；两台机器${\mathrm{中号}}_{\mathrm{一世}}$ 和 ${\mathrm{中号}}_{{\mathrm{一世}}^{\prime }}$ 具有相同的机器类型，如果 $p_{\mathrm{一世}，Ĵ}=p_{{\mathrm{一世}}^{\prime }，Ĵ}$ 对所有人 $Ĵ\in Ĵ$。此结果推断出存在用于统一并行机的PTAS（$问|b\mathrm{一种}G|C_{米\mathrm{一种}X}$）机器速度和袋数均恒定时。然后，我们提出一个$b/2$图平衡问题的-近似算法 $b\ge 2$袋; 近似值对于$b=3$，除非 $\mathsf{P}=\mathsf{NP}$，并且该算法解决了图平衡问题 $b=2$袋在多项式时间内。此外，针对所有作业均具有单位长度的带袋的并行并行机，我们提出了多项式时间算法，用于约束分配问题。为了补充我们的算法结果，我们表明，当作业的长度为1或2时，$\mathsf{NP}$-对于受限分配和图平衡问题，很难在小于3/2的因数内逼近最佳制造期 $b=2$ 袋和 $b=3$袋。我们还证明$问|b\mathrm{一种}G|C_{米\mathrm{一种}X}$ 与 $b=2$ 包包很坚固 $\mathsf{NP}$-硬。