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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 [...]

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]

We study statistical calibration, i.e., adjusting features of a computational model that are not observable or controllable in its associated physical system. We focus on functional calibration, which arises in many manufacturing processes where the unobservable features, called calibration variables, are a function of the input variables. A major challenge in many applications is that computational models are expensive and can only be evaluated a limited number of times. Furthermore, without making strong assumptions, the calibration variables are not identifiable. We propose Bayesian Non-isometric Matching Calibration (BNMC) that allows calibration of expensive computational models with only a limited number of samples taken from a computational model and its associated physical system. BNMC replaces the computational model with a dynamic Gaussian process whose parameters are trained in the calibration procedure. To resolve the identifiability issue, we present the calibration problem from a geometric perspective of non-isometric curve to surface matching, which enables us to take advantage of combinatorial optimization techniques to extract necessary information for constructing prior distributions. Our numerical experiments demonstrate that in terms of prediction accuracy BNMC outperforms, or is comparable to, other existing calibration frameworks. [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]

Background: High-throughput sequencing technology has revolutionized both medical and biological research by generating exceedingly large numbers of genetic variants. The resulting datasets share a number of common characteristics that might lead to poor generalization capacity. Concerns include noise accumulated due to the large number of predictors, sparse information regarding the p≫n problem, and overfitting and model mis-identification resulting from spurious collinearity. Additionally, complex correlation patterns are present among variables. As a consequence, reliable variable selection techniques play a pivotal role in predictive analysis, generalization capability, and robustness in clustering, as well as interpretability of the derived models. Methods and findings: K-dominating set, a parameterized graph-theoretic generalization model, was used to model SNP (single nucleotide polymorphism) data as a similarity network and searched for representative SNP variables. In particular, each SNP was represented as a vertex in the graph, (dis)similarity measures such as correlation coefficients or pairwise linkage disequilibrium were estimated to describe the relationship between each pair of SNPs; a pair of vertices are adjacent, i.e. joined by an edge, if the pairwise similarity measure exceeds a user-specified threshold. A minimum k-dominating set in the SNP graph was then made as the smallest subset such that every SNP that is excluded from the subset has at least k neighbors in the selected ones. The strength of k-dominating set selection in identifying independent variables, and in culling representative variables that are highly correlated with others, was demonstrated by a simulated dataset. The advantages of k-dominating set variable selection were also illustrated in two applications: pedigree reconstruction using SNP profiles of 1,372 Douglas-fir trees, and species delineation for 226 grasshopper mouse samples. A C++ source code that implements SNP-SELECT and uses Gurobi optimization solver for the k-dominating set variable selection is available (). [ABSTRACT FROM AUTHOR]

A k-plex is a clique relaxation introduced in social network analysis to model cohesive social subgroups that allows for a limited number of nonadjacent vertices (strangers) inside the cohesive subgroup. Several exact algorithms and heuristic approaches to find a maximum-size k-plex in the graph have been developed recently for this NP-hard problem. This article develops a greedy randomized adaptive search procedure (GRASP) for the maximum k-plex problem. We offer a key improvement in the design of the construction procedure that alleviates a drawback observed in multiple past studies. In existing construction heuristics, k-plexes found for smaller values of parameter k are sometimes not found for larger k even though they are feasible; instead inferior solutions are found. We identify the reasons behind this behavior and address these in our new construction procedure. We then show that an existing exact algorithm for solving this problem on power-law graphs can be considerably enhanced by using GRASP. The overall approach is able to solve the problem to optimality on massive social networks, including some with several million vertices and edges. These are orders of magnitude larger than the largest real-life social networks on which this problem has been solved to optimality in the current literature. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 69(4), 388-407 2017 [ABSTRACT FROM AUTHOR]

A method for constructing test functions for global optimization which utilizes continuous formulations of combinatorial optimization problems is suggested. In particular, global optimization formulations for the maximum independent set, maximum clique, and MAX CUT problems on arbitrary graphs are considered, and proofs for some of them are given. A number of sample test functions based on these formulations are proposed. [ABSTRACT FROM AUTHOR]