Improved algorithmic results for unsplittable stable allocation problems

The stable allocation problem is a many-to-many generalization of the well-known stable marriage problem, where we seek a bipartite assignment between, say, jobs (of varying sizes) and machines (of varying capacities) that is “stable” based on a set of underlying preference lists submitted by the jobs and machines. Building on the initial work of Dean et al. (The unsplittable stable marriage problem, 2006), we study a natural “unsplittable” variant of this problem, where each assigned job must be fully assigned to a single machine. Such unsplittable bipartite assignment problems generally tend to be NP-hard, including previously-proposed variants of the unsplittable stable allocation problem (McDermid and Manlove in J Comb Optim 19(3): 279–303, 2010). Our main result is to show that under an alternative model of stability, the unsplittable stable allocation problem becomes solvable in polynomial time; although this model is less likely to admit feasible solutions than the model proposed in McDermid and Manlove (J Comb Optim 19(3): 279–303, McDermid and Manlove 2010), we show that in the event there is no feasible solution, our approach computes a solution of minimal total congestion (overfilling of all machines collectively beyond their capacities). We also describe a technique for rounding the solution of a stable allocation problem to produce “relaxed” unsplit solutions that are only mildly infeasible, where each machine is overcongested by at most a single job.