Cycle Selections

We introduce the following cycle selection problem which is motivated by an application to kidney exchange problems. Given a directed graph, a cycle selection is a subset of arcs forming a union of directed cycles. A related optimization problem, the Maximum Weighted Cycle Selection problem can be defined as follows: given a weight for each arc, find a cycle selection which maximizes the total weight. We prove that this problem is strongly NP-hard. Next, we focus on cycle selections in complete directed graphs. We provide several ILP formulations of the problem: an arc formulation featuring an exponential number of constraints which can be separated in polynomial time, four extended compact formulations, and an extended non compact formulation. We investigate the relative strength of these formulations. We concentrate on the arc formulation and on the description of the associated cycle selection polytope. We prove that this polytope is full-dimensional, and that all the inequalities used in the arc formulation are facet-defining. Furthermore, we describe three new classes of facet-defining inequalities and a class of valid inequalities. We also consider the consequences of including additional constraints on the cardinality of a selection or on the length of the associated cycles.