2,080 research outputs found

    Piecewise Temperleyan dimers and a multiple SLE8_8

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    We consider the dimer model on piecewise Temperleyan, simply connected domains, on families of graphs which include the square lattice as well as superposition graphs. We focus on the spanning tree Tδ\mathcal{T}_\delta associated to this model via Temperley's bijection, which turns out to be a Uniform Spanning Tree with singular alternating boundary conditions. Generalising the work of the second author with Peltola and Wu \cite{LiuPeltolaWuUST} we obtain a scaling limit result for Tδ\mathcal{T}_\delta. For instance, in the simplest nontrivial case, the limit of Tδ\mathcal{T}_\delta is described by a pair of trees whose Peano curves are shown to converge jointly to a multiple SLE8_8 pair. The interface between the trees is shown to be given by an SLE2(1,,1)_2(-1, \ldots, -1) curve. More generally we provide an equivalent description of the scaling limit in terms of imaginary geometry. This allows us to make use of the results developed by the first author and Laslier and Ray \cite{BLRdimers}. We deduce that, universally across these classes of graphs, the corresponding height function converges to a multiple of the Gaussian free field with boundary conditions that jump at each non-Temperleyan corner. After centering, this generalises a result of Russkikh \cite{RusskikhDimers} who proved it in the case of the square lattice. Along the way, we obtain results of independent interest on chordal hypergeometric SLE8_8; for instance we show its law is equal to that of an SLE8(ρˉ)_8 (\bar \rho) for a certain vector of force points, conditional on its hitting distribution on a specified boundary arc.Comment: 42 page

    Spectral Sparsification for Communication-Efficient Collaborative Rotation and Translation Estimation

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    We propose fast and communication-efficient optimization algorithms for multi-robot rotation averaging and translation estimation problems that arise from collaborative simultaneous localization and mapping (SLAM), structure-from-motion (SfM), and camera network localization applications. Our methods are based on theoretical relations between the Hessians of the underlying Riemannian optimization problems and the Laplacians of suitably weighted graphs. We leverage these results to design a collaborative solver in which robots coordinate with a central server to perform approximate second-order optimization, by solving a Laplacian system at each iteration. Crucially, our algorithms permit robots to employ spectral sparsification to sparsify intermediate dense matrices before communication, and hence provide a mechanism to trade off accuracy with communication efficiency with provable guarantees. We perform rigorous theoretical analysis of our methods and prove that they enjoy (local) linear rate of convergence. Furthermore, we show that our methods can be combined with graduated non-convexity to achieve outlier-robust estimation. Extensive experiments on real-world SLAM and SfM scenarios demonstrate the superior convergence rate and communication efficiency of our methods.Comment: Revised extended technical report (37 pages, 15 figures, 6 tables

    A Unifying Framework for Differentially Private Sums under Continual Observation

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    We study the problem of maintaining a differentially private decaying sum under continual observation. We give a unifying framework and an efficient algorithm for this problem for \emph{any sufficiently smooth} function. Our algorithm is the first differentially private algorithm that does not have a multiplicative error for polynomially-decaying weights. Our algorithm improves on all prior works on differentially private decaying sums under continual observation and recovers exactly the additive error for the special case of continual counting from Henzinger et al. (SODA 2023) as a corollary. Our algorithm is a variant of the factorization mechanism whose error depends on the γ2\gamma_2 and γF\gamma_F norm of the underlying matrix. We give a constructive proof for an almost exact upper bound on the γ2\gamma_2 and γF\gamma_F norm and an almost tight lower bound on the γ2\gamma_2 norm for a large class of lower-triangular matrices. This is the first non-trivial lower bound for lower-triangular matrices whose non-zero entries are not all the same. It includes matrices for all continual decaying sums problems, resulting in an upper bound on the additive error of any differentially private decaying sums algorithm under continual observation. We also explore some implications of our result in discrepancy theory and operator algebra. Given the importance of the γ2\gamma_2 norm in computer science and the extensive work in mathematics, we believe our result will have further applications.Comment: 32 page

    The galaxy of Coxeter groups

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    In this paper we introduce the galaxy of Coxeter groups -- an infinite dimensional, locally finite, ranked simplicial complex which captures isomorphisms between Coxeter systems. In doing so, we would like to suggest a new framework to study the isomorphism problem for Coxeter groups. We prove some structural results about this space, provide a full characterization in small ranks and propose many questions. In addition we survey known tools, results and conjectures. Along the way we show profinite rigidity of triangle Coxeter groups -- a result which is possibly of independent interest.Comment: 30 pages, 6 figures. v2: Incorporated referee's suggestions; Corrected a mistake in the proof of Theorem 4.25 (formerly 4.24), improved other proofs and text; Final version, to appear in the Journal of Algebr

    Topology of Cut Complexes of Graphs

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    We define the kk-cut complex of a graph GG with vertex set V(G)V(G) to be the simplicial complex whose facets are the complements of sets of size kk in V(G)V(G) inducing disconnected subgraphs of GG. This generalizes the Alexander dual of a graph complex studied by Fr\"oberg (1990), and Eagon and Reiner (1998). We describe the effect of various graph operations on the cut complex, and study its shellability, homotopy type and homology for various families of graphs, including trees, cycles, complete multipartite graphs, and the prism Kn×K2K_n \times K_2, using techniques from algebraic topology, discrete Morse theory and equivariant poset topology.Comment: 36 pages, 10 figures, 1 table, Extended Abstract accepted for FPSAC2023 (Davis

    Thick Forests

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    We consider classes of graphs, which we call thick graphs, that have their vertices replaced by cliques and their edges replaced by bipartite graphs. In particular, we consider the case of thick forests, which are a subclass of perfect graphs. We show that this class can be recognised in polynomial time, and examine the complexity of counting independent sets and colourings for graphs in the class. We consider some extensions of our results to thick graphs beyond thick forests.Comment: 40 pages, 19 figure

    Clique‐width: Harnessing the power of atoms

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    Many NP-complete graph problems are polynomial-time solvable on graph classes of bounded clique-width. Several of these problems are polynomial-time solvable on a hereditary graph class if they are so on the atoms (graphs with no clique cut-set) of . Hence, we initiate a systematic study into boundedness of clique-width of atoms of hereditary graph classes. A graph is -free if is not an induced subgraph of , and it is -free if it is both -free and -free. A class of -free graphs has bounded clique-width if and only if its atoms have this property. This is no longer true for -free graphs, as evidenced by one known example. We prove the existence of another such pair and classify the boundedness of clique-width on -free atoms for all but 18 cases

    Maximal Chordal Subgraphs

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    A chordal graph is a graph with no induced cycles of length at least 44. Let f(n,m)f(n,m) be the maximal integer such that every graph with nn vertices and mm edges has a chordal subgraph with at least f(n,m)f(n,m) edges. In 1985 Erd\H{o}s and Laskar posed the problem of estimating f(n,m)f(n,m). In the late '80s, Erd\H{o}s, Gy\'arf\'as, Ordman and Zalcstein determined the value of f(n,n2/4+1)f(n,n^2/4+1) and made a conjecture on the value of f(n,n2/3+1)f(n,n^2/3+1). In this paper we prove this conjecture and answer the question of Erd\H{o}s and Laskar, determining f(n,m)f(n,m) asymptotically for all mm and exactly for mn2/3+1m \leq n^2/3+1

    Planar Disjoint Paths, Treewidth, and Kernels

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    In the Planar Disjoint Paths problem, one is given an undirected planar graph with a set of kk vertex pairs (si,ti)(s_i,t_i) and the task is to find kk pairwise vertex-disjoint paths such that the ii-th path connects sis_i to tit_i. We study the problem through the lens of kernelization, aiming at efficiently reducing the input size in terms of a parameter. We show that Planar Disjoint Paths does not admit a polynomial kernel when parameterized by kk unless coNP \subseteq NP/poly, resolving an open problem by [Bodlaender, Thomass{\'e}, Yeo, ESA'09]. Moreover, we rule out the existence of a polynomial Turing kernel unless the WK-hierarchy collapses. Our reduction carries over to the setting of edge-disjoint paths, where the kernelization status remained open even in general graphs. On the positive side, we present a polynomial kernel for Planar Disjoint Paths parameterized by k+twk + tw, where twtw denotes the treewidth of the input graph. As a consequence of both our results, we rule out the possibility of a polynomial-time (Turing) treewidth reduction to tw=kO(1)tw= k^{O(1)} under the same assumptions. To the best of our knowledge, this is the first hardness result of this kind. Finally, combining our kernel with the known techniques [Adler, Kolliopoulos, Krause, Lokshtanov, Saurabh, Thilikos, JCTB'17; Schrijver, SICOMP'94] yields an alternative (and arguably simpler) proof that Planar Disjoint Paths can be solved in time 2O(k2)nO(1)2^{O(k^2)}\cdot n^{O(1)}, matching the result of [Lokshtanov, Misra, Pilipczuk, Saurabh, Zehavi, STOC'20].Comment: To appear at FOCS'23, 82 pages, 30 figure

    Massive SLE4_4 and the scaling limit of the massive harmonic explorer

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    The massive harmonic explorer is a model of random discrete path on the hexagonal lattice that was proposed by Makarov and Smirnov as a massive perturbation of the harmonic explorer. They argued that the scaling limit of the massive harmonic explorer in a bounded domain is a massive version of chordal SLE4_4, called massive SLE4_4, which is conformally covariant and absolutely continuous with respect to chordal SLE4_4. In this paper, we provide a full and rigorous proof of this statement. Moreover, we show that a massive SLE4_4 curve can be coupled with a massive Gaussian free field as its level line, when the field has appropriate boundary conditions
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