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Register a new userA Matrix Trickle-Down Theorem on Simplicial Complexes and Applications to Sampling Colorings
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Added by
Dorna Abdolazimi
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We show that the natural Glauber dynamics mixes rapidly and generates a random proper edge-coloring of a graph with maximum degree $Delta$ whenever the number of colors is at least $qgeq (frac{10}{3} + epsilon)Delta$, where $epsilon>0$ is arbitrary and the maximum degree satisfies $Delta geq C$ for a constant $C = C(epsilon)$ depending only on $epsilon$. For edge-colorings, this improves upon prior work cite{Vig99, CDMPP19} which show rapid mixing when $qgeq (frac{11}{3}-epsilon_0 ) Delta$, where $epsilon_0 approx 10^{-5}$ is a small fixed constant. At the heart of our proof, we establish a matrix trickle-down theorem, generalizing Oppenheims influential result, as a new technique to prove that a high dimensional simplical complex is a local spectral expander.
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We present a randomized algorithm that takes as input an undirected $n$-vertex graph $G$ with maximum degree $Delta$ and an integer $k > 3Delta$, and returns a random proper $k$-coloring of $G$. The distribution of the coloring is emph{perfectly} uniform over the set of all proper $k$-colorings; the expected running time of the algorithm is $mathrm{poly}(k,n)=widetilde{O}(nDelta^2cdot log(k))$. This improves upon a result of Huber~(STOC 1998) who obtained a polynomial time perfect sampling algorithm for $k>Delta^2+2Delta$. Prior to our work, no algorithm with expected running time $mathrm{poly}(k,n)$ was known to guarantee perfectly sampling with sub-quadratic number of colors in general. Our algorithm (like several other perfect sampling algorithms including Hubers) is based on the Coupling from the Past method. Inspired by the emph{bounding chain} approach, pioneered independently by Huber~(STOC 1998) and Haggstrom & Nelander~(Scand.{} J.{} Statist., 1999), we employ a novel bounding chain to derive our result for the graph coloring problem.
We study the problem of allocating $m$ items to $n$ agents subject to maximizing the Nash social welfare (NSW) objective. We write a novel convex programming relaxation for this problem, and we show that a simple randomized rounding algorithm gives a $1/e$ approximation factor of the objective. Our main technical contribution is an extension of Gurvitss lower bound on the coefficient of the square-free monomial of a degree $m$-homogeneous stable polynomial on $m$ variables to all homogeneous polynomials. We use this extension to analyze the expected welfare of the allocation returned by our randomized rounding algorithm.
Focusing on coupling between edges, we generalize the relationship between the normalized graph Laplacian and random walks on graphs by devising an appropriate normalization for the Hodge Laplacian -- the generalization of the graph Laplacian for simplicial complexes -- and relate this to a random walk on edges. Importantly, these random walks are intimately connected to the topology of the simplicial complex, just as random walks on graphs are related to the topology of the graph. This serves as a foundational step towards incorporating Laplacian-based analytics for higher-order interactions. We demonstrate how to use these dynamics for data analytics that extract information about the edge-space of a simplicial complex that complements and extends graph-based analysis. Specifically, we use our normalized Hodge Laplacian to derive spectral embeddings for examining trajectory data of ocean drifters near Madagascar and also develop a generalization of personalized PageRank for the edge-space of simplicial complexes to analyze a book co-purchasing dataset.
Random abstract simplicial complex representation provides a mathematical description of wireless networks and their topology. In order to reduce the energy consumption in this type of network, we intend to reduce the number of network nodes without modifying neither the connectivity nor the coverage of the network. In this paper, we present a reduction algorithm that lower the number of points of an abstract simplicial complex in an optimal order while maintaining its topology. Then, we study the complexity of such an algorithm for a network simulated by a binomial point process and represented by a Vietoris-Rips complex.
Topological Coding consists of two different kinds of mathematics: topological structure and mathematical relation. The colorings and labelings of graph theory are main techniques in topological coding applied in asymmetric encryption system. Topsnut-gpws (also, colored graphs) have the following advantages: (1) Run fast in communication networks because they are saved in computer by popular matrices rather than pictures. (2) Produce easily text-based (number-based) strings for encrypt files. (3) Diversity of asymmetric ciphers, one public-key corresponds to more private-keys, or more public-keys correspond more private-keys. (4) Irreversibility, Topsnut-gpws can generate quickly text-based (number-based) strings with bytes as long as desired, but these strings can not reconstruct the original Topsnut-gpws. (5) Computational security, since there are many non-polynomial (NP-complete, NP-hard) algorithms in creating Topsnut-gpws. (6) Provable security, since there are many mathematical conjectures (open problems) in graph labelings and graph colorings. We are committed to create more kinds of new Topsnut-gpws to approximate practical applications and antiquantum computation, and try to use algebraic method and Topsnut-gpws to establish graphic group, graphic lattice, graph homomorphism etc.
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