Cliques and their generalizations are frequently used to model “tightly knit” clusters in graphs and identifying such clusters is a popular technique used in graph-based data mining. One such model is the s-club, which is a vertex subset that induces a subgraph of diameter at most s. This model has found use in a variety of fields because low-diameter clusters have practical significance in many applications. As this property is not hereditary on vertex-induced subgraphs, the diameter of a subgraph could increase upon the removal of some vertices and the subgraph could even become disconnected. For example, star graphs have diameter two but can be disconnected by removing the central vertex. The pursuit of a fault-tolerant extension of the s-club model has spawned two variants that we study in this article: robust s-clubs and hereditary s-clubs. We analyze the complexity of the verification and optimization problems associated with these variants. Then, we propose cut-like integer programming formulations for both variants whenever possible and investigate the separation complexity of the cut-like constraints. We demonstrate through our extensive computational experiments that the algorithmic ideas we introduce enable us to solve the problems to optimality on benchmark instances with several thousand vertices. This work lays the foundations for effective mathematical programming approaches for finding fault-tolerant s-clubs in large-scale networks. History: Accepted by David Alderson, Area Editor for Network Optimization: Algorithms & Applications. Funding: The computing for this project was performed at the High Performance Computing Center at Oklahoma State University supported in part through the National Science Foundation [Grant OAC-1531128]. This material is based upon work supported by the National Science Foundation under [Grants 1662757 and 1942065]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.1231 .

In critical node problems, the task is to identify a small subset of so-called critical nodes whose deletion maximally degrades a network’s “connectivity” (however that is measured). Problems of this type have been widely studied, for example, for limiting the spread of infectious diseases. However, existing approaches for solving them have typically been limited to networks having fewer than 1,000 nodes. In this paper, we consider a variant of this problem in which the task is to delete b nodes so as to minimize the number of node pairs that remain connected by a path of length at most k. With the techniques developed in this paper, instances with up to 17,000 nodes can be solved exactly. We introduce two integer programming formulations for this problem (thin and path-like) and compare them with an existing recursive formulation. Although the thin formulation generally has an exponential number of constraints, it admits an efficient separation routine. Also helpful is a new, more general preprocessing procedure that, on average, fixes three times as many variables than before. Summary of Contribution: In this paper, we consider a distance-based variant of the critical node problem in which the task is to delete b nodes so as to minimize the number of node pairs that remain connected by a path of length at most k. This problem is motivated by applications in social networks, telecommunications, and transportation networks. In our paper, we aim to solve large-scale instances of this problem. Standard out-of-the-box approaches are unable to solve such instances, requiring new integer programming models, methodological contributions, and other computational insights. For example, we propose an algorithm for finding a maximum independent set of simplicial nodes that runs in time O(nm) that we use in a preprocessing procedure; we also prove that the separation problem associated with one of our integer programming models is NP-hard. We apply our branch-and-cut implementation to real-life networks from a variety of domains and observe speedups over previous approaches.

In the article “A linear‐size zero‐one programming model for the minimum spanning tree problem in planar graphs” (Networks 39(1) (2002), 53‐60), Williams introduced an extended formulation for the spanning tree polytope of a planar graph. This formulation is remarkably small (using only O(n) variables and constraints) and remarkably strong (defining an integral polytope). In this note, we point out that Williams' formulation, as originally stated, is incorrect. Specifically, we construct a binary feasible solution to Williams' formulation that does not represent a spanning tree. Fortunately, there is a simple fix, which is to restrict the choice of the root vertices in the primal and dual spanning trees, whereas Williams explicitly allowed them to be chosen arbitrarily. The same flaw and fix apply to a subsequent formulation of Williams (“A zero‐one programming model for contiguous land acquisition.” Geographical Analysis 34(4) (2002), 330‐349).

This article considers the node‐weighted Steiner tree (NWST) problem and the maximum‐weight connected subgraph (MWCS) problem, which have applications in the design of telecommunication networks and the analysis of biological networks. Exact algorithms with provable worst‐case runtimes are provided. The first algorithm for NWST runs in time for n‐vertex instances when the number of terminals is bounded. It is based on dynamic programming and generalizes a Steiner tree algorithm of Dreyfus and Wagner. When used alongside Hakimi's spanning tree enumeration algorithm, it implies a time algorithm for NWST. It is also shown that Hakimi's 46‐year‐old algorithm for Steiner tree is essentially best‐possible under the strong exponential time hypothesis (SETH). Then two algorithms for MWCS are provided. Their runtimes are polynomial in the number of vertices of the graph, but exponential in the number of vertices that have positive (or negative) weight. The latter is shown to be essentially best‐possible under SETH. Together, they imply that MWCS can be solved in time . To the best of the authors’ knowledge, these are the first improvements over exhaustive search in the literature.

In this article, a heuristic is said to be provably best if, assuming , no other heuristic always finds a better solution (when one exists). This extends the usual notion of “best possible” approximation algorithms to include a larger class of heuristics. We illustrate the idea on several problems that are somewhat stylized versions of real‐life network optimization problems, including the maximum clique, maximum k‐club, minimum (connected) dominating set, and minimum vertex coloring problems. The corresponding provably best construction heuristics resemble those commonly used within popular metaheuristics. Along the way, we show that it is hard to recognize whether the clique number and the k‐club number of a graph are equal, yet a polynomial‐time computable function is “sandwiched” between them. This is similar to the celebrated Lovász function wherein an efficiently computable function lies between two graph invariants that are ‐hard to compute. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 67(3), 238–245 2016

This paper considers the minimum k-connected d-dominating set problem, which is a fault-tolerant generalization of the minimum connected dominating set (MCDS) problem. Three integer programming formulations based on vertex cuts are proposed (depending on whether d < k, d = k, or d > k) and their integer hulls are studied. The separation problem for the vertex-cut inequalities is a weighted vertex-connectivity problem and is polytime solvable, meaning that the LP relaxation can be solved in polytime despite having exponentially many constraints. A new class of valid inequalities—r-robust vertex-cut inequalities—is introduced and is shown to induce exponentially many facets. Finally, a lazy-constraint approach is shown to compare favorably with existing approaches for the MCDS problem (the case k = d = 1), and is in fact the fastest in literature for standard test instances. A key subroutine is an algorithm for finding an inclusion-wise minimal vertex cut in linear time. Computational results for (k, d) = (2,1), (2,2), (3,3), (4,4) are provided as well.

This paper explores techniques for solving the maximum clique and vertex coloring problems on very large-scale real-life networks. Because of the size of such networks and the intractability of the considered problems, previously developed exact algorithms may not be directly applicable. The proposed approaches aim to reduce the network instances to a size that is tractable for existing solvers, while preserving optimality. Two clique relaxation structures are exploited for this purpose. In addition to the known k-core structure, a newly introduced clique relaxation, k-community, is used to further reduce the instance size. Experimental results on real-life graphs (collaboration networks, P2P networks, social networks, etc.) show the proposed procedures to be effective by finding, for the first time, exact solutions for instances with over 18 million vertices.