358 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    A rooted variant of Stanley's chromatic symmetric function

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    Richard Stanley defined the chromatic symmetric function XGX_G of a graph GG and asked whether there are non-isomorphic trees TT and UU with XT=XUX_T=X_U. We study variants of the chromatic symmetric function for rooted graphs, where we require the root vertex to either use or avoid a specified color. We present combinatorial identities and recursions satisfied by these rooted chromatic polynomials, explain their relation to pointed chromatic functions and rooted UU-polynomials, and prove three main theorems. First, for all non-empty connected graphs GG, Stanley's polynomial XG(x1,
,xN)X_G(x_1,\ldots,x_N) is irreducible in Q[x1,
,xN]\mathbb{Q}[x_1,\ldots,x_N] for all large enough NN. The same result holds for our rooted variant where the root node must avoid a specified color. We prove irreducibility by a novel combinatorial application of Eisenstein's Criterion. Second, we prove the rooted version of Stanley's Conjecture: two rooted trees are isomorphic as rooted graphs if and only if their rooted chromatic polynomials are equal. In fact, we prove that a one-variable specialization of the rooted chromatic polynomial (obtained by setting x0=x1=qx_0=x_1=q, x2=x3=1x_2=x_3=1, and xn=0x_n=0 for n>3n>3) already distinguishes rooted trees. Third, we answer a question of Pawlowski by providing a combinatorial interpretation of the monomial expansion of pointed chromatic functions.Comment: 21 pages; v2: added a short algebraic proof to Theorem 2 (now Theorem 15), we also answer a question of Pawlowski about monomial expansions; v3: added additional one-variable specialization results, simplified main proof

    Gallai's path decomposition conjecture for cartesian product of graphs (\uppercase\expandafter{\romannumeral 2})

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    Let GG be a graph of order nn. A path decomposition P\mathcal{P} of GG is a collection of edge-disjoint paths that covers all the edges of GG. Let p(G)p(G) denote the minimum number of paths needed in a path decomposition of GG. Gallai conjectured that if GG is connected, then p(G)≀⌈n2⌉p(G)\leq \lceil\frac{n}{2}\rceil. In this paper, we prove that Gallai's path decomposition conjecture holds for the cartesian product G□HG\Box H, where HH is any graph and GG is a unicyclic graph or a bicyclic graph.Comment: 26 pages, 2 figures. arXiv admin note: text overlap with arXiv:2310.1118

    Exactly soluble models in many-body physics

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    Almost all phenomena in the universe are described, at the fundamental level, by quantum manybody models. In general, however, a complete understanding of large systems with many degrees of freedom is impossible. While in general many-body quantum systems are intractable, there are special cases for which there are techniques that allow for an exact solution. Exactly soluble models are interesting because they are soluble; beyond this, they can be used to gain intuition for further reaching many-body systems, including when they can be leveraged to help with numerical approximations for general models. The work presented in this thesis considers exactly soluble models of quantum many-body systems. The first part of this thesis extends the family of many-body spin models for which we can find a freefermion solution. A solution method that was developed for a specific free-fermion model is generalized in such a way that allows application to a broader class of many-body spin system than was previously known to be free. Models which admit a solution via this method are characterized by a graph theory invariants: in brief it is shown that a quantum spin system has an exact description via non-interacting fermions if its frustration graph is claw-free and contains a simplicial clique. The second part of this thesis gives an explicit example of how the usefulness of exactly soluble models can extend beyond the solution itself. This chapter pertains to the calculation of the topological entanglement entropy in topologically ordered loop-gas states. Topological entanglement entropy gives an understanding of how correlations may extend throughout a system. In this chapter the topological entanglement entropy of two- and three-dimensional loop-gas states is calculated in the bulk and at the boundary. We obtain a closed form expression for the topological entanglement in terms of the anyonic theory that the models support

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization

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    Many real-life decision problems are discrete in nature. To solve such problems as mathematical optimization problems, integrality constraints are commonly incorporated in the model to reflect the choice of finitely many alternatives. At the same time, it is known that semidefinite programming is very suitable for obtaining strong relaxations of combinatorial optimization problems. In this dissertation, we study the interplay between semidefinite programming and integrality, where a special focus is put on the use of cutting-plane methods. Although the notions of integrality and cutting planes are well-studied in linear programming, integer semidefinite programs (ISDPs) are considered only recently. We show that manycombinatorial optimization problems can be modeled as ISDPs. Several theoretical concepts, such as the ChvĂĄtal-Gomory closure, total dual integrality and integer Lagrangian duality, are studied for the case of integer semidefinite programming. On the practical side, we introduce an improved branch-and-cut approach for ISDPs and a cutting-plane augmented Lagrangian method for solving semidefinite programs with a large number of cutting planes. Throughout the thesis, we apply our results to a wide range of combinatorial optimization problems, among which the quadratic cycle cover problem, the quadratic traveling salesman problem and the graph partition problem. Our approaches lead to novel, strong and efficient solution strategies for these problems, with the potential to be extended to other problem classes

    Root and weight semigroup rings for signed posets

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    We consider a pair of semigroups associated to a signed poset, called the root semigroup and the weight semigroup, and their semigroup rings, RPrtR_P^\mathrm{rt} and RPwtR_P^\mathrm{wt}, respectively. Theorem 4.1.5 gives generators for the toric ideal of affine semigroup rings associated to signed posets and, more generally, oriented signed graphs. These are the subrings of Laurent polynomials generated by monomials of the form ti±1,ti±2,ti±1tj±1t_i^{\pm 1},t_i^{\pm 2},t_i^{\pm 1}t_j^{\pm 1}. This result appears to be new and generalizes work of Boussicault, F\'eray, Lascoux and Reiner, of Gitler, Reyes, and Villarreal, and of Villarreal. Theorem 4.2.12 shows that strongly planar signed posets PP have rings RPrtR_P^\mathrm{rt}, RP√R_{P^{\scriptscriptstyle\vee}} which are complete intersections, with Corollary 4.2.20 showing how to compute ΚP\Psi_P in this case. Theorem 5.2.3 gives a Gr\"obner basis for the toric ideal of RPwtR_P^{\mathrm{wt}} in type B, generalizing Proposition 6.4 of F\'eray and Reiner. Theorems 5.3.10 and 5.3.1 give two characterizations (via forbidden subposets versus via inductive constructions) of the situation where this Gr\"obner basis gives a complete intersection presentation for its initial ideal, generalizing Theorems 10.5 and 10.6 of F\'eray and Reiner.Comment: 170 pages; 63 figures; PhD Dissertation, University of Minnesota, August 201

    Visualized Algorithm Engineering on Two Graph Partitioning Problems

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    Concepts of graph theory are frequently used by computer scientists as abstractions when modeling a problem. Partitioning a graph (or a network) into smaller parts is one of the fundamental algorithmic operations that plays a key role in classifying and clustering. Since the early 1970s, graph partitioning rapidly expanded for applications in wide areas. It applies in both engineering applications, as well as research. Current technology generates massive data (“Big Data”) from business interactions and social exchanges, so high-performance algorithms of partitioning graphs are a critical need. This dissertation presents engineering models for two graph partitioning problems arising from completely different applications, computer networks and arithmetic. The design, analysis, implementation, optimization, and experimental evaluation of these models employ visualization in all aspects. Visualization indicates the performance of the implementation of each Algorithm Engineering work, and also helps to analyze and explore new algorithms to solve the problems. We term this research method as “Visualized Algorithm Engineering (VAE)” to emphasize the contribution of the visualizations in these works. The techniques discussed here apply to a broad area of problems: computer networks, social networks, arithmetic, computer graphics and software engineering. Common terminologies accepted across these disciplines have been used in this dissertation to guarantee practitioners from all fields can understand the concepts we introduce

    Beyond Flatland : exploring graphs in many dimensions

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    Societies, technologies, economies, ecosystems, organisms, . . . Our world is composed of complex networks—systems with many elements that interact in nontrivial ways. Graphs are natural models of these systems, and scientists have made tremendous progress in developing tools for their analysis. However, research has long focused on relatively simple graph representations and problem specifications, often discarding valuable real-world information in the process. In recent years, the limitations of this approach have become increasingly apparent, but we are just starting to comprehend how more intricate data representations and problem formulations might benefit our understanding of relational phenomena. Against this background, our thesis sets out to explore graphs in five dimensions: descriptivity, multiplicity, complexity, expressivity, and responsibility. Leveraging tools from graph theory, information theory, probability theory, geometry, and topology, we develop methods to (1) descriptively compare individual graphs, (2) characterize similarities and differences between groups of multiple graphs, (3) critically assess the complexity of relational data representations and their associated scientific culture, (4) extract expressive features from and for hypergraphs, and (5) responsibly mitigate the risks induced by graph-structured content recommendations. Thus, our thesis is naturally situated at the intersection of graph mining, graph learning, and network analysis.Gesellschaften, Technologien, Volkswirtschaften, Ökosysteme, Organismen, . . . Unsere Welt besteht aus komplexen Netzwerken—Systemen mit vielen Elementen, die auf nichttriviale Weise interagieren. Graphen sind natĂŒrliche Modelle dieser Systeme, und die Wissenschaft hat bei der Entwicklung von Methoden zu ihrer Analyse große Fortschritte gemacht. Allerdings hat sich die Forschung lange auf relativ einfache GraphreprĂ€sentationen und Problemspezifikationen beschrĂ€nkt, oft unter VernachlĂ€ssigung wertvoller Informationen aus der realen Welt. In den vergangenen Jahren sind die Grenzen dieser Herangehensweise zunehmend deutlich geworden, aber wir beginnen gerade erst zu erfassen, wie unser VerstĂ€ndnis relationaler PhĂ€nomene von intrikateren DatenreprĂ€sentationen und Problemstellungen profitieren kann. Vor diesem Hintergrund erkundet unsere Dissertation Graphen in fĂŒnf Dimensionen: DeskriptivitĂ€t, MultiplizitĂ€t, KomplexitĂ€t, ExpressivitĂ€t, und Verantwortung. Mithilfe von Graphentheorie, Informationstheorie, Wahrscheinlichkeitstheorie, Geometrie und Topologie entwickeln wir Methoden, welche (1) einzelne Graphen deskriptiv vergleichen, (2) Gemeinsamkeiten und Unterschiede zwischen Gruppen multipler Graphen charakterisieren, (3) die KomplexitĂ€t relationaler DatenreprĂ€sentationen und der mit ihnen verbundenen Wissenschaftskultur kritisch beleuchten, (4) expressive Merkmale von und fĂŒr Hypergraphen extrahieren, und (5) verantwortungsvoll den Risiken begegnen, welche die Graphstruktur von Inhaltsempfehlungen mit sich bringt. Damit liegt unsere Dissertation naturgemĂ€ĂŸ an der Schnittstelle zwischen Graph Mining, Graph Learning und Netzwerkanalyse
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