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

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 \(4/3\) -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 \(O(m\log (nUA)\cdot \operatorname{MF}(kn,m+k^2n))\) time, where \(n\) is the number of nodes, \(m\) is the number of edges, \(k\) is the number of terminals, \(A\) is the maximum edge-cost, \(U\) is the maximum edge-capacity, and \(\operatorname{MF}(n',m')\) is the time complexity of a max-flow algorithm in a network with \(n'\) nodes and \(m'\) edges. The algorithm implies that the \(4/3\) -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.