Completing partial schedules for open shop with unit processing times and routing

Open Shop is a classical scheduling problem: given a set J of jobs and a set M of machines, find a minimum-makespan schedule to process each job J_i ∈ J on each machine M_q ∈ 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 setup times. Routing Open Shop is NP-hard for |V | = |M| = 2. For the special case with unit processing times piq = 1, we exploit Galvin’s theorem about listcoloring 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 2^{O(|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.