9 research outputs found

    Arrangements of equal minors in the positive

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    Abstract. We discuss arrangements of equal minors in totally positive matrices. More precisely, we would like to investigate the structure of possible equalities and inequalities between the minors. We show that arrangements of equals minors of largest value are in bijection with sorted sets, which earlier appeared in the context of alcoved polytopes and Gröbner bases. Maximal arrangements of this form correspond to simplices of the alcoved triangulation of the hypersimplex; and the number of such arrangements equals the Eulerian number. On the other hand, we conjecture and prove in many cases that arrangements of equal minors of smallest value are exactly the weakly separated sets. Weakly separated sets, originally introduced by Leclerc and Zelevinsky, are closely related to the positive Grassmannian and the associated cluster algebra. RĂ©sumĂ©. Il s’agit des arrangements des mineurs Ă©gaux dans les matrices totalement positives. Plus prĂ©cisĂ©ment, nous aimerions Ă©tudier la structure des Ă©galitĂ©s et inĂ©galitĂ©s possibles entre les mineurs. Nous montrons que les arrangements des mineurs Ă©gaux de plus grande valeur sont en bijection avec les ensembles triĂ©s, qui auparavant apparaissaient dans le cadre de polytopes alcĂŽve et bases de Gröbner. Arrangements maximales de ce format correspondent aux simplexes de la triangulation alcĂŽve de la hypersimplex, et le nombre de ces arrangements est Ă©gal au nombre eulĂ©rien. D’autre part, nous conjecturons et prouvons dans des cas nombreux que les arrangements des mineurs Ă©gaux de plus petite valeur sont notamment les ensembles faiblement sĂ©parĂ©s. Ces ensembles faiblement sĂ©parĂ©s, initialement introduites par Leclerc et Zelevinsky, sont liĂ©s Ă  la Grassmannienne positive et l’algĂšbre cluster

    On the Maximum Crossing Number

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    Research about crossings is typically about minimization. In this paper, we consider \emph{maximizing} the number of crossings over all possible ways to draw a given graph in the plane. Alpert et al. [Electron. J. Combin., 2009] conjectured that any graph has a \emph{convex} straight-line drawing, e.g., a drawing with vertices in convex position, that maximizes the number of edge crossings. We disprove this conjecture by constructing a planar graph on twelve vertices that allows a non-convex drawing with more crossings than any convex one. Bald et al. [Proc. COCOON, 2016] showed that it is NP-hard to compute the maximum number of crossings of a geometric graph and that the weighted geometric case is NP-hard to approximate. We strengthen these results by showing hardness of approximation even for the unweighted geometric case and prove that the unweighted topological case is NP-hard.Comment: 16 pages, 5 figure

    Arrangements of equal minors in the positive Grassmannian

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    We discuss arrangements of equal minors in totally positive matrices. More precisely, we would like to investigate the structure of possible equalities and inequalities between the minors. We show that arrangements of equals minors of largest value are in bijection with sorted sets, which earlier appeared in the context of alcoved polytopes and Gröbner bases. Maximal arrangements of this form correspond to simplices of the alcoved triangulation of the hypersimplex; and the number of such arrangements equals the Eulerian number. On the other hand, we conjecture and prove in many cases that arrangements of equal minors of smallest value are exactly the weakly separated sets. Weakly separated sets, originally introduced by Leclerc and Zelevinsky, are closely related to the \textitpositive Grassmannian and the associated cluster algebra

    Computing crossing numbers

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    The graph theoretic problem of crossing numbers has been around for over 60 years, but still very little is known about this simple, yet intricate nonplanarity measure. The question is easy to state: Given a graph, draw it in the plane with the minimum number of edge crossings. A lot of research has been devoted to giving an answer to this question, not only by graph theoreticians, but also by computer scientists. The crossing number is central to areas like chip design and automatic graph drawing. While there are algorithms to solve the problem heuristically, we know that it is in general NP-complete. Furthermore, we do not know if the problem is efficiently approximable, except for some special cases. In this thesis, we tackle the problem using Mathematical Programming. We show how to formulate the crossing number problem as systems of linear inequalities, and discuss how to solve these formulations for reasonably sized graphs to provable optimality in acceptable time--despite its theoretical complexity class. We present non-standard branch-and-cut-and-price techniques to achieve this goal, and introduce an efficient preprocessing algorithm, also valid for other traditional non-planarity measures. We discuss extensions of these ideas to related crossing number variants arising in practice, and show a practical application of a formerly purely theoretic crossing number derivative. The thesis also contains an extensive experimental study of the formulations and algorithms presented herein, and an outlook on its applicability for graph theoretic questions regarding the crossing numbers of special graph classes.Das Kreuzungszahlproblem wird von Graphentheoretikern seit ĂŒber 60 Jahren betrachtet, jedoch ist noch immer sehr wenig ĂŒber dieses einfache und zugleich hochkomplizierte Ma der NichtplanaritĂ€t bekannt. Die Aufgabenstellung ist simpel: Gegeben ein Graph, zeichnen Sie ihn mit der kleinstmöglichen Anzahl an Kantenkreuzungen. Nicht nur Graphentheoretiker sondern auch Informatiker beschĂ€ftigten sich ausgiebig mit dieser Aufgabe, denn es handelt sich dabei um ein zentrales Konzept im Chipdesign und im automatischen Graphenzeichnen. Zwar existieren Algorithmen um das Problem heuristisch zu lösen, jedoch wissen wir, dass es im Allgemeinen NP-vollstĂ€ndig ist. DarĂŒberhinaus ist auch unbekannt, ob sich das Problem, außer in SpezialfĂ€llen, effizient approximieren lĂ€sst. In dieser Dissertation, versuchen wir das Problem mit Hilfe der Mathematischen Programmierung zu lösen. Wir zeigen, wie man das Kreuzungszahlproblem als verschiedene Systeme von linearen Ungleichungen formulieren kann und diskutieren wie man diese Formulierungen fĂŒr nicht allzu große Graphen beweisbar optimal und in akzeptabler Zeit lösen kann - unabhĂ€ngig von seiner formalen KomplexitĂ€tsklasse. Wir stellen dazu benötigte maßgeschneiderte Branch-and-Cut-and-Price Techniken vor, und prĂ€sentieren einen effizienten Algorithmus zur Vorverarbeitung; dieser ist auch fĂŒr andere traditionelle Ma e der NichtplanaritĂ€t geeignet. Wir diskutieren Erweiterungen unserer Ideen fĂŒr verwandte Kreuzungszahlkonzepte die in der Praxis auftreten, und zeigen eine praktische Anwendung eines vormals rein theoretisch behandelten Kreuzungszahl-Derivats auf. Diese Arbeit enthĂ€lt auch eine ausfĂŒhrliche experimentelle Studie der prĂ€sentierten Formulierungen und Algorithmen, sowie einen Ausblick ĂŒber deren mögliche Nutzung fĂŒr graphentheoretische Fragen bezĂŒglich der Kreuzungszahlen von speziellen Graphenklassen
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