Increasing reliability and reducing disruptions in supply networks are of increasing importance; for example, power outages in electricity distribution networks cost \$35-50 billion annually in the US. Motivated by the operational constraints of such networks and their rapid adoption of decentralized paradigms and self-healing components, we introduce the "minimum reconnection time" (MRT) problem. MRT seeks to reduce outage time after network disruptions by programming reconnection times of different edges (i.e., switches), while ensuring that the operating network is acyclic. We show that MRT is NP-hard and is a special case of the well-known minimum linear ordering problem (MLOP) in the submodular optimization literature. MLOP is a special case of a broader class of ordering problems that often admit polynomial time approximation algorithms. We develop the theory of kernel-based randomized rounding approaches to give a tight polynomial-time approximation for MRT, improving the state-of-the-art approximation factor for a broad class of MLOP instances. Further, motivated by the reliability incentive structure for utility companies and operational energy losses in distribution networks, we propose local search over spanning trees to balance multiple objectives simultaneously. We computationally validate our reconfiguration methods on the NREL SMART-DS Greensboro synthetic network, and show that this improves service equity by a factor of four, across industrial and residential areas.