A fixed-parameter algorithm for a routing open shop problem: unit processing times, few machines and locations

Open Shop is a classical scheduling problem: given a set J of jobs and a setM of machines, find a minimum-makespan schedule to process each job Ji ∈ J on each machine Mq ∈ M for a given amount piq 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 graph 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 or batch setup times. Routing Open Shop is NP-hard for |V | = |M| = 2. For the special case with unit processing times piq = 1, we exploit a variant of 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 2O(|V ||M| 2 log |V ||M|) · poly(|J|) time, that is, fixed-parameter tractable parameterized by |V | + |M|. Various upper bounds shown using the schedule completion theorem suggest it to be likewise beneficial for the development of approximation algorithms.