Protecting the Connectivity of a Graph Under Nonuniform Edge Failures

Abstract. We study the problem of guaranteeing the connectivity of a given graph by protecting or strengthening edges. Herein, a protected edge is assumed to be robust and will not fail, which features a nonuniform failure model. We introduce the [Formula: see text]-Steiner-Connectivity Preservation problem where we protect a minimum-cost set of edges such that the underlying graph maintains [Formula: see text]-edge-connectivity between given terminal pairs against edge failures, assuming at most [Formula: see text] unprotected edges can fail. We design polynomial-time exact algorithms for the cases where [Formula: see text] and [Formula: see text] are small and approximation algorithms for general values of [Formula: see text] and [Formula: see text]. Additionally, we show that when both [Formula: see text] and [Formula: see text] are part of the input, even deciding whether a given solution is feasible is [Formula: see text]-complete. This hardness also carries over to Flexible Network Design, a research direction that has gained significant attention. In particular, previous work focuses on problem settings where either [Formula: see text] or [Formula: see text] is constant, for which our new hardness result now provides justification.