Given an undirected graph $G=(V,E)$ (i.e. the conflict graph) where $V$ is a set of $n$ vertices (representing the jobs), processing times $p \colon V \to \mathbb{Z}_>$, and $m\geq 2$ identical machines the Parallel Machine Scheduling with Conflicts (PMC) consists in finding an assignment $c \colon V \to [m]:=\{1,\ldots, m\}$ with $c(u)\neq c(v)$ for all $\{u,v\} \in E$ that minimizes the makespan $\max_{k \in [m]} \sum_{v \in V \colon c(v)=k} p(v)$. First we consider the natural assignment formulation for PMC using binary variables indexed by the jobs and machines, and discuss how to reduce the symmetries in such model. Then we propose a compact mixed integer linear programming formulation for PMC to tackle the issues related to symmetry and unbalanced enumeration tree associated with the assignment model. The proposed formulation for PMC uses a set of representative jobs (one in each machine) to express feasible solutions of the problem, and it is based on the representatives model for the vertex coloring problem. We present a polyhedral study of the associated polytope, and show classes of valid inequalities inherited from the stable set polytope. We describe branch-and-cut algorithms for PMC, and report on preliminary computational experiments with benchmark instances.

翻译：给定无向图 $G=(V,E)$（即冲突图），其中 $V$ 为 $n$ 个顶点（代表作业）的集合，加工时间 $p \colon V \to \mathbb{Z}_>$，以及 $m\geq 2$ 台相同机器，并行机器带冲突调度问题 (PMC) 旨在寻找一个赋值 $c \colon V \to [m]:=\{1,\ldots, m\}$，满足对所有 $\{u,v\} \in E$ 有 $c(u)\neq c(v)$，同时最小化最大完工时间 $\max_{k \in [m]} \sum_{v \in V \colon c(v)=k} p(v)$。首先，我们考虑使用由作业和机器索引的二元变量构建的 PMC 自然赋值形式化描述，并讨论如何减少该模型中的对称性。随后，针对赋值模型中的对称性及不平衡枚举树问题，我们提出一种紧致的 PMC 混合整数线性规划形式化描述。所提形式化描述通过选取一组代表性作业（每台机器选取一个）来表达问题的可行解，其基础源于顶点着色问题的代表元模型。我们开展关联多面体的多面体研究，并展示从稳定集多面体继承的有效不等式类。最后，描述 PMC 的分支切割算法，并报告基于基准实例的初步计算实验结果。