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

Authors Hiroshi Hirai , Motoki Ikeda

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Hiroshi Hirai
  • Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, Japan
Motoki Ikeda
  • Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, Japan


We thank the referees for helpful comments.

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Hiroshi Hirai and Motoki Ikeda. Node-Connectivity Terminal Backup, Separately-Capacitated Multiflow, and Discrete Convexity. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 65:1-65:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


The terminal backup problems [Anshelevich and Karagiozova, 2011] 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 (2016) gave a 4/3-approximation algorithm based on 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 (mUA)⋅ MF(kn,m+k²n)) time, where 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(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, called a separately-capacitated multiflow. We show a min-max theorem which extends Lovász - Cherkassky theorem to the node-capacity setting. Our results build on discrete convex analysis for the node-connectivity terminal backup problem.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial optimization
  • terminal backup problem
  • node-connectivity
  • separately-capacitated multiflow
  • discrete convex analysis


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