三台均匀机上具有三次不相容图的单位长度工件排序
Scheduling of unit-length jobs with cubic incompatibility graphs on three uniform machines
摘要：
本文研究了在速度为$s_1, s_2,$和速度为$s_3$的3台均匀机上调度$n$相同工件的问题，使调度长度最小。我们假设工作受到某种互斥约束，建模的立方不相容图。我们证明了如果图是2-色的，那么问题可以在$O(n^2)$时间内解决。如果图是3-色的，则即使$s_1>s_2=s_3$也是NP-难的。然而，在这种情况下，存在以$O(n^3)$时间运行的$4/3$近似算法。而且，当$3s_1/4 \leq s_2 = s_3$满足条件时，该算法几乎必然最优地解决了问题.
英文摘要：
In the paper we consider the problem of scheduling $n$ identical jobs on 3 uniform machines with speeds $s_1, s_2,$ and $s_3$ to minimize the schedule length. We assume that jobs are subjected to some kind of mutual exclusion constraints, modeled by a cubic incompatibility graph. We show that if the graph is 2-chromatic then the problem can be solved in $O(n^2)$ time. If the graph is 3-chromatic, the problem becomes NP-hard even if $s_1>s_2=s_3$. However, in this case there exists a $4/3$-approximation algorithm running in $O(n^3)$ time. Moreover, this algorithm solves the problem almost surely to optimality if $3s_1/4 \leq s_2 = s_3$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
