Search

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]

This paper introduces the variable objective search framework for combinatorial optimization. The method utilizes different objective functions used in alternative mathematical programming formulations of the same combinatorial optimization problem in an attempt to improve the solutions obtained using each of these formulations individually. The proposed technique is illustrated using alternative quadratic unconstrained binary formulations of the classical maximum independent set problem in graphs. [ABSTRACT FROM AUTHOR]

This article investigates the local maxima properties of a box-constrained quadratic optimization formulation of the maximum independent set problem in graphs. Theoretical results characterizing binary local maxima in terms of certain induced subgraphs of the given graph are developed. We also consider relations between continuous local maxima of the quadratic formulation and binary local maxima in the Hamming distance-1 and distance-2 neighborhoods. These results are then used to develop an efficient local search algorithm that provides considerable speed-up over a typical local search algorithm for the binary Hamming distance-2 neighborhood. [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]