Journal of Combinatorial Optimization

Caughman, John S.; Dunn, Charles L.; Laison, Joshua D.; Neudauer, Nancy Ann; Starr, Colin L.

2023 Journal of Combinatorial Optimization

doi: 10.1007/s10878-023-01084-9

A rectangle visibility graph (RVG) is represented by assigning to each vertex a rectangle in the plane with horizontal and vertical sides in such a way that edges in the graph correspond to unobstructed horizontal and vertical lines of sight between their corresponding rectangles. To discretize, we consider only rectangles whose corners have integer coordinates. For any given RVG, we seek a representation with smallest bounding box as measured by its area, perimeter, height, or width (height is assumed not to exceed width). We derive a number of results regarding these parameters. Using these results, we show that these four measures are distinct, in the sense that there exist graphs G1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$G_1$$\end{document} and G2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$G_2$$\end{document} with area(G1)<area(G2)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\,\textrm{area}\,}}(G_1) < {{\,\textrm{area}\,}}(G_2)$$\end{document} but perim(G2)<perim(G1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\,\textrm{perim}\,}}(G_2) < {{\,\textrm{perim}\,}}(G_1)$$\end{document}, and analogously for all other pairs of these parameters. We further show that there exists a graph G3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$G_3$$\end{document} with representations S1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$S_1$$\end{document} and S2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$S_2$$\end{document} such that area(G3)=area(S1)<area(S2)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\,\textrm{area}\,}}(G_3)={{\,\textrm{area}\,}}(S_1)<{{\,\textrm{area}\,}}(S_2)$$\end{document} but perim(G3)=perim(S2)<perim(S1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\,\textrm{perim}\,}}(G_3)={{\,\textrm{perim}\,}}(S_2)<{{\,\textrm{perim}\,}}(S_1)$$\end{document}. In other words, G3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$G_3$$\end{document} requires distinct representations to minimize area and perimeter. Similarly, such graphs exist to demonstrate the independence of all other pairs of these parameters. Among graphs with n≤6\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n \le 6$$\end{document} vertices, the empty graph En\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$E_n$$\end{document} requires largest area. But for graphs with n=7\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n=7$$\end{document} and n=8\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n=8$$\end{document} vertices, we show that the complete graphs K7\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K_7$$\end{document} and K8\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K_8$$\end{document} require larger area than E7\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$E_7$$\end{document} and E8\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$E_8$$\end{document}, respectively. Using this, we show that for all n≥8\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n \ge 8$$\end{document}, the empty graph En\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$E_n$$\end{document} does not have largest area, perimeter, height, or width among all RVGs on n vertices.

Gu, Haisheng; Li, Qing; Shen, Duanming

2023 Journal of Combinatorial Optimization

doi: 10.1007/s10878-023-01085-8

With the rapid development of information technology, a large number of sensitive words have appeared, which has brought great harm to network security and social stability. Therefore, how to identify and classify these sensitive information accurately has become an important issue. Combined with the improved artificial network algorithm, an adaptive classification algorithm is proposed in the experiment, which can provide local intelligent classification service according to the classification results. At the same time, the algorithm transforms the clustering model structure of the traditional network algorithm, introduces the classification information, so that it can be applied to the classification problem, and expands the application field of the data field theory. The experimental results show that: (1) From the experimental results of different text quantities, the SOM algorithm assigns the classification task to different levels of nodes, and realizes the modularization of detection. (2) The overall mean results show that the highest recall rate is 87%, which has met the basic grading criteria, and the detection accuracy of sensitive words will also be improved. (3) The experimental results show that the algorithm can accurately classify the sensitive and speed up the parameter optimization, and is superior to the comparison algorithm in many indicators. (4) From the simulation results, compared with the traditional neural network algorithm, the precision and recall of the algorithm are maintained at more than 90% and the loss is less than 0.11.

Escobedo, Adolfo R.; Yasmin, Romena

2023 Journal of Combinatorial Optimization

doi: 10.1007/s10878-023-01075-w

Ordering polytopes have been instrumental to the study of combinatorial optimization problems arising in a variety of fields including comparative probability, computational social choice, and group decision-making. The weak order polytope is defined as the convex hull of the characteristic vectors of all binary orders on n alternatives that are reflexive, transitive, and total. By and large, facet defining inequalities (FDIs) of this polytope have been obtained through simple enumeration and through connections with other combinatorial polytopes. This paper derives five new large classes of FDIs by utilizing the equivalent representations of a weak order as a ranking of n alternatives that allows ties; this connection simplifies the construction of valid inequalities, and it enables groupings of characteristic vectors into useful structures. We demonstrate that a number of FDIs previously obtained through enumeration are actually special cases of the large classes. This work also introduces novel construction procedures for generating affinely independent members of the identified ranking structures. Additionally, it states two conjectures on how to derive many more large classes of FDIs using the featured techniques.