Critical edges for the assignment problem : complexity and exact resolution

Cristina Bazgan (1) , Sónia Toubaline (1) , Daniel Vanderpooten (1)

1 LAMSADE - Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision

This paper investigates two problems related to the determination of critical edges for the minimum cost assignment problem. Given a complete bipartite balanced graph with nn vertices on each part and with costs on its edges, kkMost Vital Edges Assignment consists of determining a set of kk edges whose removal results in the largest increase in the cost of a minimum cost assignment. A dual problem, Min Edge Blocker Assignment, consists of removing a subset of edges of minimum cardinality such that the cost of a minimum cost assignment in the remaining graph is larger than or equal to a specified threshold. We show that kkMost Vital Edges Assignment is NPNP-hard to approximate within a factor c<2c<2 and Min Edge Blocker Assignment is NPNP-hard to approximate within a factor 1.361.36. We also provide an exact algorithm for kkMost Vital Edges Assignment that runs in O(nk+2)O(nk+2). This algorithm can also be used to solve exactly Min Edge Blocker Assignment.

Mots clés: Assignment problem; Approximation; Complexity; Exact algorithm; Min edge blocker; Most vital edges

Domaines: Informatique [cs]

Cristina Bazgan, Sónia Toubaline, Daniel Vanderpooten. Critical edges for the assignment problem : complexity and exact resolution. Operations Research Letters, 2013, 41 (6), ⟨10.1016/j.orl.2013.10.001⟩. ⟨hal-01499696⟩