A polyhedral approach to a constrained assignment problem

Conference · Sat Dec 31 04:00:00 EST 1994

OSTI ID:36120

Hefner, A; Kleinschmidt, P

We consider the following problem (called the Master/Slave-Matching Problem): Given an undirected bipartite graph G(V, E) with bipartition V = W {union} U and a digraph D = (U, A). A Master/Slave-matching in G is a matching in G such that for every arc (u, v) {element_of} A the node v is matched whenever the node u is matched. The problem is, to find a Master/Slave-matching of maximum cardinality. The motivation for the study of this problem arises from the area of manpower scheduling where one tries to assign workers to jobs such that the assigned jobs satisfy certain dependence constraints. Let k be the maximum size of a (weakly) connected component of D. First we show that the problem is NP-hard and remains NP-hard even if k = 3. Second we focus on the case k = 2: We use polyhedral combinatorics to establish a min-max equation which well-characterizes the cardinality of a maximum Master/Slave-matching. This equation can be viewed as a generalization of Konig`s min-max theorem. Finally, we show (for k = 2) how the weighted Master/Slave-Matching Problem (where every edge e {element_of} E has a nonnegative weight) can be transformed to the (non-bipartite) weighted Matching Problem.

• OSTI ID: 36120
• Report Number(s): CONF-9408161--
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
ALGORITHMS
MATRICES
NONLINEAR PROBLEMS
NONLINEAR PROGRAMMING
NUMERICAL SOLUTION