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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. [ABSTRACT FROM AUTHOR]

A γ-quasi-clique in a simple undirected graph refers to a subset of vertices that induces a subgraph with edge density at least γ. When γ equals one, this definition corresponds to a classical clique. When γ is less than one, it relaxes the requirement of all possible edges by the clique definition. Quasi-clique detection has been used in graph-based data mining to find dense clusters, especially in large-scale error-prone data sets in which the clique model can be overly restrictive. The maximum γ-quasi-clique problem, seeking a γ-quasi-clique of maximum cardinality in the given graph, can be formulated as an optimization problem with a linear objective function and a single quadratic constraint in binary variables. This article investigates the Lagrangian dual of this formulation and develops an upper-bounding technique using the geometry of ellipsoids to bound the Lagrangian dual. The tightness of the upper bound is compared with those obtained from multiple mixed-integer programming formulations of the problem via experiments on benchmark instances. [ABSTRACT FROM AUTHOR]

Detecting large 2-clubs in biological, social and financial networks can help reveal important information about the structure of the underlying systems. In large-scale networks that are error-prone, the uncertainty associated with the existence of an edge between two vertices can be modeled by assigning a failure probability to that edge. Here, we study the problem of detecting large 'risk-averse' 2-clubs in graphs subject to probabilistic edge failures. To achieve risk aversion, we first model the loss in 2-club property due to probabilistic edge failures as a function of the decision (chosen 2-club cluster) and randomness (graph structure). Then, we utilize the conditional value-at-risk (CVaR) of the loss for a given decision as a quantitative measure of risk for that decision, which is bounded in the model. More precisely, the problem is modeled as a CVaR-constrained single-stage stochastic program. The main contribution of this article is a new Benders decomposition algorithm that outperforms an existing decomposition approach on a test-bed of randomly generated instances, and real-life biological and social networks. [ABSTRACT FROM AUTHOR]

Detecting quasi-cliques in graphs is a useful tool for detecting dense clusters in graph-based data mining. Particularly in large-scale data sets that are error-prone, cliques are overly restrictive and impractical. Quasi-clique detection has been accomplished using heuristic approaches in various applications of graph-based data mining in protein interaction networks, gene co-expression networks, and telecommunication networks. Quasi-cliques are not hereditary, in the sense that every subset of a quasi-clique need not be a quasi-clique. This lack of heredity introduces interesting challenges in the development of exact algorithms to detect maximum cardinality quasi-cliques. The only exact approaches for this problem are limited to two mixed integer programming formulations that were recently proposed in the literature. The main contribution of this article is a new combinatorial branch-and-bound algorithm for the maximum quasi-clique problem. [ABSTRACT FROM AUTHOR]

In this paper, we describe an application of prescriptive analytics to enhance data-driven decision making at a specialty steel bar products supplier and manufacturer in North America. As part of the company's daily business, it must make available-to-promise (ATP) decisions, which determine in real time the dates by which it can promise delivery of products that customers requested during the quotation stage. Previously, a salesperson had to make such decisions by analyzing reports on available inventory. To support these ATP decisions, we developed a real-time decision support system (DSS) to find an optimal assignment of the available inventory and to support additional what-if analysis. The DSS uses a suite of mixed-integer programming models and commercial software to solve the models. The company has incorporated the DSS into its enterprise resource planning system to seamlessly facilitate its use of business analytics. [ABSTRACT FROM AUTHOR]

Given a simple undirected graph, the problem of finding a maximum subset of vertices satisfying a nontrivial, interesting property Π that is hereditary on induced subgraphs, is known to be NP-hard. Many well-known graph properties meet the above conditions, making the problem widely applicable. This paper proposes a general purpose exact algorithmic framework to solve this problem and investigates key algorithm design and implementation issues that are helpful in tailoring the general framework for specific graph properties. The performance of the algorithms so derived for the maximum s- plex and the maximum s- defective clique problems, which arise in network-based data mining applications, is assessed through a computational study. [ABSTRACT FROM AUTHOR]

This paper introduces and studies the maximum k-plex problem, which arises in social network analysis and has wider applicability in several important areas employing graph-based data mining. After establishing NP-completeness of the decision version of the problem on arbitrary graphs, an integer programming formulation is presented, followed by a polyhedral study to identify combinatorial valid inequalities and facets. A branch-and-cut algorithm is implemented and tested on proposed benchmark instances. An algorithmic approach is developed exploiting the graph-theoretic properties of a k-plex that is effective in solving the problem to optimality on very large, sparse graphs such as the power law graphs frequently encountered in the applications of interest. [ABSTRACT FROM AUTHOR]