2,129 research outputs found

    Asymptotics of lattice walks via analytic combinatorics in several variables

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    We consider the enumeration of walks on the two dimensional non-negative integer lattice with short steps. Up to isomorphism there are 79 unique two dimensional models to consider, and previous work in this area has used the kernel method, along with a rigorous computer algebra approach, to show that 23 of the 79 models admit D-finite generating functions. In 2009, Bostan and Kauers used Pad\'e-Hermite approximants to guess differential equations which these 23 generating functions satisfy, in the process guessing asymptotics of their coefficient sequences. In this article we provide, for the first time, a complete rigorous verification of these guesses. Our technique is to use the kernel method to express 19 of the 23 generating functions as diagonals of tri-variate rational functions and apply the methods of analytic combinatorics in several variables (the remaining 4 models have algebraic generating functions and can thus be handled by univariate techniques). This approach also shows the link between combinatorial properties of the models and features of its asymptotics such as asymptotic and polynomial growth factors. In addition, we give expressions for the number of walks returning to the x-axis, the y-axis, and the origin, proving recently conjectured asymptotics of Bostan, Chyzak, van Hoeij, Kauers, and Pech.Comment: 10 pages, 3 tables, as accepted to proceedings of FPSAC 2016 (without conference formatting

    Parametric shortest-path algorithms via tropical geometry

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    We study parameterized versions of classical algorithms for computing shortest-path trees. This is most easily expressed in terms of tropical geometry. Applications include shortest paths in traffic networks with variable link travel times.Comment: 24 pages and 8 figure

    Antimagic Labelings of Caterpillars

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    A kk-antimagic labeling of a graph GG is an injection from E(G)E(G) to {1,2,…,∣E(G)∣+k}\{1,2,\dots,|E(G)|+k\} such that all vertex sums are pairwise distinct, where the vertex sum at vertex uu is the sum of the labels assigned to edges incident to uu. We call a graph kk-antimagic when it has a kk-antimagic labeling, and antimagic when it is 0-antimagic. Hartsfield and Ringel conjectured that every simple connected graph other than K2K_2 is antimagic, but the conjecture is still open even for trees. Here we study kk-antimagic labelings of caterpillars, which are defined as trees the removal of whose leaves produces a path, called its spine. As a general result, we use constructive techniques to prove that any caterpillar of order nn is (⌊(n−1)/2⌋−2)(\lfloor (n-1)/2 \rfloor - 2)-antimagic. Furthermore, if CC is a caterpillar with a spine of order ss, we prove that when CC has at least ⌊(3s+1)/2⌋\lfloor (3s+1)/2 \rfloor leaves or ⌊(s−1)/2⌋\lfloor (s-1)/2 \rfloor consecutive vertices of degree at most 2 at one end of a longest path, then CC is antimagic. As a consequence of a result by Wong and Zhu, we also prove that if pp is a prime number, any caterpillar with a spine of order pp, p−1p-1 or p−2p-2 is 11-antimagic.Comment: 13 pages, 4 figure

    Spin chain simulations with a meron cluster algorithm

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    We apply a meron cluster algorithm to the XY spin chain, which describes a quantum rotor. This is a multi-cluster simulation supplemented by an improved estimator, which deals with objects of half-integer topological charge. This method is powerful enough to provide precise results for the model with a theta-term - it is therefore one of the rare examples, where a system with a complex action can be solved numerically. In particular we measure the correlation length, as well as the topological and magnetic susceptibility. We discuss the algorithmic efficiency in view of the critical slowing down. Due to the excellent performance that we observe, it is strongly motivated to work on new applications of meron cluster algorithms in higher dimensions.Comment: 18 pages, 9 figures, published versio

    Generating Random Elements of Finite Distributive Lattices

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    This survey article describes a method for choosing uniformly at random from any finite set whose objects can be viewed as constituting a distributive lattice. The method is based on ideas of the author and David Wilson for using ``coupling from the past'' to remove initialization bias from Monte Carlo randomization. The article describes several applications to specific kinds of combinatorial objects such as tilings, constrained lattice paths, and alternating-sign matrices.Comment: 13 page

    A Tight Lower Bound for Counting Hamiltonian Cycles via Matrix Rank

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    For even kk, the matchings connectivity matrix Mk\mathbf{M}_k encodes which pairs of perfect matchings on kk vertices form a single cycle. Cygan et al. (STOC 2013) showed that the rank of Mk\mathbf{M}_k over Z2\mathbb{Z}_2 is Θ(2k)\Theta(\sqrt 2^k) and used this to give an O∗((2+2)pw)O^*((2+\sqrt{2})^{\mathsf{pw}}) time algorithm for counting Hamiltonian cycles modulo 22 on graphs of pathwidth pw\mathsf{pw}. The same authors complemented their algorithm by an essentially tight lower bound under the Strong Exponential Time Hypothesis (SETH). This bound crucially relied on a large permutation submatrix within Mk\mathbf{M}_k, which enabled a "pattern propagation" commonly used in previous related lower bounds, as initiated by Lokshtanov et al. (SODA 2011). We present a new technique for a similar pattern propagation when only a black-box lower bound on the asymptotic rank of Mk\mathbf{M}_k is given; no stronger structural insights such as the existence of large permutation submatrices in Mk\mathbf{M}_k are needed. Given appropriate rank bounds, our technique yields lower bounds for counting Hamiltonian cycles (also modulo fixed primes pp) parameterized by pathwidth. To apply this technique, we prove that the rank of Mk\mathbf{M}_k over the rationals is 4k/poly(k)4^k / \mathrm{poly}(k). We also show that the rank of Mk\mathbf{M}_k over Zp\mathbb{Z}_p is Ω(1.97k)\Omega(1.97^k) for any prime p≠2p\neq 2 and even Ω(2.15k)\Omega(2.15^k) for some primes. As a consequence, we obtain that Hamiltonian cycles cannot be counted in time O∗((6−ϵ)pw)O^*((6-\epsilon)^{\mathsf{pw}}) for any ϵ>0\epsilon>0 unless SETH fails. This bound is tight due to a O∗(6pw)O^*(6^{\mathsf{pw}}) time algorithm by Bodlaender et al. (ICALP 2013). Under SETH, we also obtain that Hamiltonian cycles cannot be counted modulo primes p≠2p\neq 2 in time O∗(3.97pw)O^*(3.97^\mathsf{pw}), indicating that the modulus can affect the complexity in intricate ways.Comment: improved lower bounds modulo primes, improved figures, to appear in SODA 201
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