The Minimum Spanning k-Core Problem with Bounded CVaR Under Probabilistic Edge Failures

Abstract

This article introduces the minimum spanning k-core problem that seeks to find a spanning subgraph with a minimum degree of at least k (also known as a k-core) that minimizes the total cost of the edges in the subgraph. The concept of k-cores was introduced in social network analysis to identify denser portions of a social network. We exploit the graph-theoretic properties of this model to introduce a new approach to survivable interhub network design via spanning k-cores; the model preserves connectivity and diameter under limited edge failures. The deterministic version of the problem is polynomial-time solvable due to its equivalence to generalized graph matching. We propose two conditional value-at-risk (CVaR) constrained optimization models to obtain risk-averse solutions for the minimum spanning k-core problem under probabilistic edge failures. We present polyhedral reformulations of the convex piecewise linear loss functions used in these models that enable Benders-like decomposition approaches. A decomposition and branch-and-cut approach is then developed to solve the scenario-based approximation of the CVaR-constrained minimum spanning k-core problem for the aforementioned loss functions. The computational performance of the algorithm is investigated via numerical experiments.

Conditional value‐at‐risk beyond finance: a survey

“…Graphs are used to model several important practical applications that arise in different domains, which may be uncertain in nature. Examples include traffic flows over highways subject to extreme congestion that may cause a road (i.e., a link of the network) failure (see Boginski et al., ); movements of vehicles in battlefield scenarios where a path cannot be accessible due to an ongoing fire fight (see Boginski et al., ); graph models of data subject to measurement uncertainties in capturing the relationships (i.e., links) between data entries (i.e., nodes) (see Mahdavi Pajouh et al., ); networks that are constructed using data extraction processes that are potentially error‐prone (see Yezerska et al., ); communication networks where it is crucial that the communication is not interrupted if a link fails (see Ma et al., ). The main characteristics of each paper reviewed in this section are summarized in Table .…”

“…Stochastic extensions of network problems that aim at determining subgraphs that satisfy given structural properties are tackled in Ma et al. (), Mahdavi Pajouh et al. (), and Yezerska et al.…”

“…Ma et al. () study the problem of determining a minimum‐cost spanning subgraph of a random graph such that the degree of every node is at least k . The authors cast this problem as an MILP where the CVaR is employed to limit potential violations of the above‐mentioned minimum k degree requirement.…”

“…Exact algorithms are proposed by Ma et al. () (branch and cut), Mahdavi Pajouh et al. () (Benders decomposition algorithms), Rysz et al.…”

The Minimum Edge Compact Spanner Network Design Problem

“…Some of the well known network design problems are the minimum spanning tree problem [ [24]], the Steiner tree problem [ [25]], the survival network design problem (SNDP) [ [26]], the uniform buy-at-bulk network design problem [ [27]], and the traveling salesman problem [ [24]]. Most of these problems attempt to find subgraphs of an input graph with various spanning properties [ [28,29,30]]. For example, in the Steiner tree problem, we are given a graph with vertex set V and a set of terminal nodes T .…”