Cycle selections

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