Journal of Global Optimization

, Volume 74, Issue 4, pp 783–801 | Cite as

On the chance-constrained minimum spanning k-core problem

  • Juan Ma
  • Balabhaskar BalasundaramEmail author


A graph is called a k-core if every vertex has at least k neighbors. If the parameter k is sufficiently large relative to the number of vertices, a k-core is guaranteed to possess 2-hop reachability between all pairs of vertices. Furthermore, it is guaranteed to preserve those pairwise distances under arbitrary single-vertex deletion. Hence, the concept of a k-core can be used to produce 2-hop survivable network designs, specifically to design inter-hub networks. Formally, given an edge-weighted graph, the minimum spanningk-core problem seeks a spanning subgraph of the given graph that is a k-core with minimum total edge weight. For any fixed k, this problem is equivalent to a generalized graph matching problem and can be solved in polynomial time. This article focuses on a chance-constrained version of the minimum spanning k-core problem under probabilistic edge failures. We first show that this probabilistic version is NP-hard, and we conduct a polyhedral study to strengthen the formulation. The quality of bounds produced by the strengthened formulation is demonstrated through a computational study.


Chance-constrained optimization Minimum spanning k-core Hop-constrained survivable network design 



The computational experiments reported in this article were conducted at the Oklahoma State University High Performance Computing Center. The authors would also like to thank Foad Mahdavi Pajouh for helpful discussions regarding the complexity and lifting results.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Turner Broadcasting System, Inc.AtlantaUSA
  2. 2.School of Industrial Engineering and ManagementOklahoma State UniversityStillwaterUSA

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