Node-Connectivity Terminal Backup, Separately Capacitated Multiflow, and Discrete Convexity

Abstract

The terminal backup problems ([E. Anshelevich and A. Karagiozova, SIAM J. Comput., 40 (2011), pp. 678–708]) form a class of network design problems: Given an undirected graph with a requirement on terminals, the goal is to find a minimum-cost subgraph satisfying the connectivity requirement. The node-connectivity terminal backup problem requires a terminal to connect other terminals with a number of node-disjoint paths. This problem is not known whether is NP-hard or tractable. Fukunaga [SIAM J. Discrete Math., 30 (2016), pp. 777–800] gave a -approximation algorithm based on an LP-rounding scheme using a general LP-solver. In this paper, we develop a combinatorial algorithm for the relaxed LP to find a half-integral optimal solution in time, where is the number of nodes, is the number of edges, is the number of terminals, is the maximum edge-cost, is the maximum edge-capacity, and is the time complexity of a max-flow algorithm in a network with nodes and edges. The algorithm implies that the -approximation algorithm for the node-connectivity terminal backup problem is also efficiently implemented. For the design of algorithm, we explore a connection between the node-connectivity terminal backup problem and a new type of a multiflow, which is called a separately capacitated multiflow. We show a min-max theorem which extends the Lovász–Cherkassky theorem to the node-capacity setting. Our results build on discrete convexity in the node-connectivity terminal backup problem.