1. NODE-CONNECTIVITY TERMINAL BACKUP, SEPARATELY CAPACITATED MULTIFLOW, AND DISCRETE CONVEXITY.
- Author
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HIROSHI HIRAI and MOTOKI IKEDA
- Subjects
- *
UNDIRECTED graphs , *APPROXIMATION algorithms , *ALGORITHMS - 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 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(mlog(n-A · MF(/cn,m T/c2n)) 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 MF(nz,mz) is the time complexity of a max-flow algorithm in a network with nz 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 Lovasz-Cherkassky theorem to the node-capacity setting. Our results build on discrete convexity in the node-connectivity terminal backup problem. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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