Completing Partial Schedules for Open Shop with Unit Processing Times and Routing

Abstract

Open Shop is a classical scheduling problem: given a seti¾ź$$\mathcal J$$ of jobs and a seti¾ź$$\mathcal M$$ of machines, find a minimum-makespan schedule to process each jobi¾ź$$J_i\in \mathcal J$$ on each machinei¾ź$$M_q\in \mathcal M$$ for a given amounti¾ź$$p_{iq}$$ of time such that each machine processes only one job at a time and each job is processed by only one machine at a time. In Routing Open Shop, the jobs are located in the vertices of an edge-weighted graphi¾ź$$\mathcal G=V,E$$ whose edge weights determine the time needed for the machines to travel between jobs. The travel times also have a natural interpretation as sequence-dependent family setup times. Routing Open Shop is NP-hard for $$|V|=|\mathcal M|=2$$. For the special case with unit processing timesi¾ź$$p_{iq}=1$$, we exploit Galvin's theorem about list-coloring edges of bipartite graphs to prove a theorem that gives a sufficient condition for the completability of partial schedules. Exploiting this schedule completion theorem and integer linear programming, we show that Routing Open Shop with unit processing times is solvable in [InlineEquation not available: see fulltext.]i¾źtime, that is, fixed-parameter tractable parameterized byi¾ź$$|V|+|\mathcal M|$$. Various upper bounds shown using the schedule completion theorem suggest it to be likewise beneficial for the development of approximation algorithms.