Scheduling with conflicts: formulations, valid inequalities, and computational experiments
CoRR(2023)
摘要
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.
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