An [formula omitted]– [formula omitted] connection problem with adaptability

Abstract

We study a new two-stage version of an s – t path problem, which we call adaptable robust connection path (ARCP). Given an undirected graph G = ( V , E ) , two vertices s , t ∈ V and two integers f , r ∈ Z + , ARCP asks to find a set S ⊂ E of minimum cardinality which connects s and t , such that for any ‘failure set’ F ⊂ E with | F | ≤ f , the set of edges S ∖ F can be completed to a set which connects s and t by adding at most r edges from E ∖ F . We show the problem is NP-hard, and there is no polynomial-time α -approximation algorithm for the problem for α < 2 unless P=NP. For f = r = 1 we provide an exact polynomial algorithm, and for f = 1 and arbitrary r we provide a polynomial 2-approximation algorithm. A characterization of the feasible set is provided for f = 1 and several links are established between ARCP and other combinatorial optimization problems, including a new combinatorial optimization problem, the minimum reduced cost cycle problem whose complexity is open.