The hypergraph assignment problem

The hypergraph assignment problem (HAP) is the generalization of assignments from directed graphs to directed hypergraphs. It serves, in particular, as a universal tool to model several train composition rules in vehicle rotation planning for long distance passenger railways. We prove that even for problems with a small hyperarc size and hypergraphs with a special partitioned structure the HAP is NP-hard and APX-hard. Further, we present an extended integer linear programming formulation which implies, e. g., all clique inequalities. 1 Directed Hypergraph Terminology Definition 1.1. A directed hypergraph D = (V, W, A) is a pair of two disjoint vertex sets V , W of the same size |V | = |W | and a set of hyperarcs A ⊆ 2V · ∪W such that every hyperarc a ∈ A has the same size |a ∩ V | = |a ∩W | > 0 of vertices in V and W . If |a ∩ V |= |a ∩W |= 1, then we call the hyperarc a also an arc. In the usual definition of a directed hypergraph we have only one vertex set and every hyperarc is a pair of two sets called its tail and its head [GLPN93]. Despite that we do not require a nonempty intersection of the tail and the head, our definition is equivalent to it. To get this form of the directed hypergraph identify for the vertex sets V = {v1, . . . , vn}, W = {w1, . . . , wn} the vertex v1 with w1, . . . , vn with wn and take for each hyperarc a ∈ A, a ∩ V as its tail and a ∩W as its head. Definition 1.2. Let D = (V, W, A) be a directed hypergraph. A hyperassignment in D is a subset H ⊆ A of the hyperarcs such that for every v ∈ V ∪W there exists exactly one a ∈ H such that v ∈ a. For an example of a directed hypergraph and a hyperassignment see figure 1. Definition 1.3. Given a set S, a cost function is a function cS : S → R. For T ⊆ S we define cS(T ) := ∑ s∈T cS(s). ∗Zuse Institute Berlin, Takustraße 7, 14195 Berlin, Germany, [email protected] ∗∗Zuse Institute Berlin, Takustraße 7, 14195 Berlin, Germany, [email protected]