332 research outputs found

    A triangle process on graphs with given degree sequence

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    The triangle switch Markov chain is designed to generate random graphs with given degree sequence, but having more triangles than would appear under the uniform distribution. Transition probabilities of the chain depends on a parameter, called the activity, which is used to assign higher stationary probability to graphs with more triangles. In previous work we proved ergodicity of the triangle switch chain for regular graphs. Here we prove ergodicity for all sequences with minimum degree at least 3, and show rapid mixing of the chain when the activity and the maximum degree are not too large. As far as we are aware, this is the first rigorous analysis of a Markov chain algorithm for generating graphs from a a known non-uniform distribution.Comment: 35 page

    Ergodicity bounds for stable Ornstein-Uhlenbeck systems in Wasserstein distance with applications to cutoff stability

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    This article establishes cutoff stability also known as abrupt thermalization for generic multidimensional Hurwitz stable Ornstein-Uhlenbeck systems with (possibly degenerate) L\'evy noise at fixed noise intensity. The results are based on several ergodicity quantitative lower and upper bounds some of which make use of the recently established shift linearity property of the Wasserstein-Kantorovich-Rubinstein distance by the authors. It covers such irregular systems like Jacobi chains and more general networks of coupled harmonic oscillators with a heat bath (including L\'evy excitations) at constant temperature on the outer edges and the so-called Brownian gyrator.Comment: 29 pages, 3 figure

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Processing massive graphs under limited visibility

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    Graphs are one of the most important and widely used combinatorial structures in mathematics. Their ability to model many real world scenarios which involve a large network of related entities make them useful across disciplines. They are useful as an abstraction in the analysis of networked structures such as the Internet, social networks, road networks, biological networks and many more. The graphs arising out of many real world phenomenon can be very large and they keep evolving over time. For example, Facebook reported over 2:9 billion monthly active users in 2022. Another very large and dynamic network is the human brain consisting of around 1011 nodes and many more edges. These large and evolving graphs present new challenges for algorithm designers. Traditional graph algorithms designed to work with centralised and sequential computing models are rendered useless due to their prohibitively high resource usage. In fact one needs huge amounts of resources just to read the entire graph. A number of new theoretical models have been devised over the years to keep up with the trends in the modern computing systems capable of handing massive input datasets. Some of these models such as streaming model and the query model allow the algorithm to view the graph piecemeal. In some cases, the model allows the graph to be processed by a set of interconnected computing elements such as in distributed computing. In this thesis we address some graph problems in these non-centralised, non-sequential models of computing with a limited access to the input graph. Specifically, we address three different graph problems, each in a different computing model. The first problem we look at is the computation of approximate shortest paths in dynamic streams. The second problem deals with finding kings in tournament graphs, given query access to the arcs of the tournament. The third and the final problem we investigate is a local test criteria for testing the expansion of a graph in the distributed CONGEST model

    The PACE 2022 Parameterized Algorithms and Computational Experiments Challenge: Directed Feedback Vertex Set

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    The Parameterized Algorithms and Computational Experiments challenge (PACE) 2022 was devoted to engineer algorithms solving the NP-hard Directed Feedback Vertex Set (DFVS) problem. The DFVS problem is to find a minimum subset XVX ⊆ V in a given directed graph G=(V,E)G = (V,E) such that, when all vertices of XX and their adjacent edges are deleted from GG, the remainder is acyclic. Overall, the challenge had 90 participants from 26 teams, 12 countries, and 3 continents that submitted their implementations to this year’s competition. In this report, we briefly describe the setup of the challenge, the selection of benchmark instances, as well as the ranking of the participating teams. We also briefly outline the approaches used in the submitted solvers

    Applications

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    Volume 3 describes how resource-aware machine learning methods and techniques are used to successfully solve real-world problems. The book provides numerous specific application examples: in health and medicine for risk modelling, diagnosis, and treatment selection for diseases in electronics, steel production and milling for quality control during manufacturing processes in traffic, logistics for smart cities and for mobile communications

    University of Windsor Undergraduate Calendar 2023 Spring

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    https://scholar.uwindsor.ca/universitywindsorundergraduatecalendars/1023/thumbnail.jp

    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum

    Karp's patching algorithm on random perturbations of dense digraphs

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    We consider the following question. We are given a dense digraph D0D_0 with minimum in- and out-degree at least αn\alpha n, where α>0\alpha>0 is a constant. We then add random edges RR to D0D_0 to create a digraph DD. Here an edge ee is placed independently into RR with probability nϵn^{-\epsilon} where ϵ>0\epsilon>0 is a small positive constant. The edges of DD are given edge costs C(e),eE(D)C(e),e\in E(D), where C(e)C(e) is an independent copy of the exponential mean one random variable EXP(1)EXP(1) i.e. Pr(EXP(1)x)=ex\Pr(EXP(1)\geq x)=e^{-x}. Let C(i,j),i,j[n]C(i,j),i,j\in[n] be the associated n×nn\times n cost matrix where C(i,j)=C(i,j)=\infty if (i,j)E(D)(i,j)\notin E(D). We show that w.h.p. the patching algorithm of Karp finds a tour for the asymmetric traveling salesperson problem that is asymptotically equal to that of the associated assignment problem. Karp's algorithm runs in polynomial time.Comment: Fixed the proof of a lemm

    Naval Postgraduate School Academic Catalog - February 2023

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