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We introduce
$\varepsilon $
-approximate versions of the notion of a Euclidean vector bundle for
$\varepsilon \geq 0$
, which recover the classical notion of a Euclidean vector bundle when
$\varepsilon = 0$
. In particular, we study Čech cochains with coefficients in the orthogonal group that satisfy an approximate cocycle condition. We show that
$\varepsilon $
-approximate vector bundles can be used to represent classical vector bundles when
$\varepsilon> 0$
is sufficiently small. We also introduce distances between approximate vector bundles and use them to prove that sufficiently similar approximate vector bundles represent the same classical vector bundle. This gives a way of specifying vector bundles over finite simplicial complexes using a finite amount of data and also allows for some tolerance to noise when working with vector bundles in an applied setting. As an example, we prove a reconstruction theorem for vector bundles from finite samples. We give algorithms for the effective computation of low-dimensional characteristic classes of vector bundles directly from discrete and approximate representations and illustrate the usage of these algorithms with computational examples.
The problem of inverting the total divergence operator is central to finding components of a given conservation law. This might not be taxing for a low-order conservation law of a scalar partial differential equation, but integrable systems have conservation laws of arbitrarily high order that must be found with the aid of computer algebra. Even low-order conservation laws of complex systems can be hard to find and invert. This paper describes a new, efficient approach to the inversion problem. Two main tools are developed: partial Euler operators and partial scalings. These lead to a line integral formula for the inversion of a total derivative and a procedure for inverting a given total divergence concisely.
We present an efficient algorithm to generate a discrete uniform distribution on a set of p elements using a biased random source for p prime. The algorithm generalizes Von Neumann’s method and improves the computational efficiency of Dijkstra’s method. In addition, the algorithm is extended to generate a discrete uniform distribution on any finite set based on the prime factorization of integers. The average running time of the proposed algorithm is overall sublinear:
$\operatorname{O}\!(n/\log n)$
.
In this work, we prove a version of the Sylvester–Gallai theorem for quadratic polynomials that takes us one step closer to obtaining a deterministic polynomial time algorithm for testing zeroness of
$\Sigma ^{[3]}\Pi \Sigma \Pi ^{[2]}$
circuits. Specifically, we prove that, if a finite set of irreducible quadratic polynomials
${\mathcal {Q}}$
satisfies that for every two polynomials
$Q_1,Q_2\in {\mathcal {Q}}$
there is a subset
${\mathcal {K}}\subset {\mathcal {Q}}$
such that
$Q_1,Q_2 \notin {\mathcal {K}}$
and whenever
$Q_1$
and
$Q_2$
vanish, then
$\prod _{i\in {\mathcal {K}}} Q_i$
vanishes, then the linear span of the polynomials in
${\mathcal {Q}}$
has dimension
$O(1)$
. This extends the earlier result [21] that holds for the case
$|{\mathcal {K}}| = 1$
.
We prove a surprising symmetry between the law of the size
$G_n$
of the greedy independent set on a uniform Cayley tree
$ \mathcal{T}_n$
of size n and that of its complement. We show that
$G_n$
has the same law as the number of vertices at even height in
$ \mathcal{T}_n$
rooted at a uniform vertex. This enables us to compute the exact law of
$G_n$
. We also give a Markovian construction of the greedy independent set, which highlights the symmetry of
$G_n$
and whose proof uses a new Markovian exploration of rooted Cayley trees that is of independent interest.
The random-cluster model is a unifying framework for studying random graphs, spin systems and electrical networks that plays a fundamental role in designing efficient Markov Chain Monte Carlo (MCMC) sampling algorithms for the classical ferromagnetic Ising and Potts models. In this paper, we study a natural non-local Markov chain known as the Chayes–Machta (CM) dynamics for the mean-field case of the random-cluster model, where the underlying graph is the complete graph on n vertices. The random-cluster model is parametrised by an edge probability p and a cluster weight q. Our focus is on the critical regime:
$p = p_c(q)$
and
$q \in (1,2)$
, where
$p_c(q)$
is the threshold corresponding to the order–disorder phase transition of the model. We show that the mixing time of the CM dynamics is
$O({\log}\ n \cdot \log \log n)$
in this parameter regime, which reveals that the dynamics does not undergo an exponential slowdown at criticality, a surprising fact that had been predicted (but not proved) by statistical physicists. This also provides a nearly optimal bound (up to the
$\log\log n$
factor) for the mixing time of the mean-field CM dynamics in the only regime of parameters where no non-trivial bound was previously known. Our proof consists of a multi-phased coupling argument that combines several key ingredients, including a new local limit theorem, a precise bound on the maximum of symmetric random walks with varying step sizes and tailored estimates for critical random graphs. In addition, we derive an improved comparison inequality between the mixing time of the CM dynamics and that of the local Glauber dynamics on general graphs; this results in better mixing time bounds for the local dynamics in the mean-field setting.
We study the computational complexity of approximating the partition function of the ferromagnetic Ising model with the external field parameter $\lambda $ on the unit circle in the complex plane. Complex-valued parameters for the Ising model are relevant for quantum circuit computations and phase transitions in statistical physics but have also been key in the recent deterministic approximation scheme for all $|\lambda |\neq 1$ by Liu, Sinclair and Srivastava. Here, we focus on the unresolved complexity picture on the unit circle and on the tantalising question of what happens around $\lambda =1$, where, on one hand, the classical algorithm of Jerrum and Sinclair gives a randomised approximation scheme on the real axis suggesting tractability and, on the other hand, the presence of Lee–Yang zeros alludes to computational hardness. Our main result establishes a sharp computational transition at the point $\lambda =1$ and, more generally, on the entire unit circle. For an integer $\Delta \geq 3$ and edge interaction parameter $b\in (0,1)$, we show $\mathsf {\#P}$-hardness for approximating the partition function on graphs of maximum degree $\Delta $ on the arc of the unit circle where the Lee–Yang zeros are dense. This result contrasts with known approximation algorithms when $|\lambda |\neq 1$ or when $\lambda $ is in the complementary arc around $1$ of the unit circle. Our work thus gives a direct connection between the presence/absence of Lee–Yang zeros and the tractability of efficiently approximating the partition function on bounded-degree graphs.
In this paper we analyze a simple spectral method (EIG1) for the problem of matrix alignment, consisting in aligning their leading eigenvectors: given two matrices A and B, we compute two corresponding leading eigenvectors
$v_1$
and
$v'_{\!\!1}$
. The algorithm returns the permutation
$\hat{\pi}$
such that the rank of coordinate
$\hat{\pi}(i)$
in
$v_1$
and that of coordinate i in
$v'_{\!\!1}$
(up to the sign of
$v'_{\!\!1}$
) are the same.
We consider a model of weighted graphs where the adjacency matrix A belongs to the Gaussian orthogonal ensemble of size
$N \times N$
, and B is a noisy version of A where all nodes have been relabeled according to some planted permutation
$\pi$
; that is,
$B= \Pi^T (A+\sigma H) \Pi $
, where
$\Pi$
is the permutation matrix associated with
$\pi$
and H is an independent copy of A. We show the following zero–one law: with high probability, under the condition
$\sigma N^{7/6+\epsilon} \to 0$
for some
$\epsilon>0$
, EIG1 recovers all but a vanishing part of the underlying permutation
$\pi$
, whereas if
$\sigma N^{7/6-\epsilon} \to \infty$
, this method cannot recover more than o(N) correct matches.
This result gives an understanding of the simplest and fastest spectral method for matrix alignment (or complete weighted graph alignment), and involves proof methods and techniques which could be of independent interest.
Coupling-from-the-past (CFTP) methods have been used to generate perfect samples from finite Gibbs hard-sphere models, an important class of spatial point processes consisting of a set of spheres with the centers on a bounded region that are distributed as a homogeneous Poisson point process (PPP) conditioned so that spheres do not overlap with each other. We propose an alternative importance-sampling-based rejection methodology for the perfect sampling of these models. We analyze the asymptotic expected running time complexity of the proposed method when the intensity of the reference PPP increases to infinity while the (expected) sphere radius decreases to zero at varying rates. We further compare the performance of the proposed method analytically and numerically with that of a naive rejection algorithm and of popular dominated CFTP algorithms. Our analysis relies upon identifying large deviations decay rates of the non-overlapping probability of spheres whose centers are distributed as a homogeneous PPP.
We analyse the behaviour of the Euclidean algorithm applied to pairs (g,f) of univariate nonconstant polynomials over a finite field
$\mathbb{F}_{q}$
of q elements when the highest degree polynomial g is fixed. Considering all the elements f of fixed degree, we establish asymptotically optimal bounds in terms of q for the number of elements f that are relatively prime with g and for the average degree of
$\gcd(g,f)$
. We also exhibit asymptotically optimal bounds for the average-case complexity of the Euclidean algorithm applied to pairs (g,f) as above.
We consider the problem of computing the partition function
$\sum _x e^{f(x)}$
, where
$f: \{-1, 1\}^n \longrightarrow {\mathbb R}$
is a quadratic or cubic polynomial on the Boolean cube
$\{-1, 1\}^n$
. In the case of a quadratic polynomial f, we show that the partition function can be approximated within relative error
$0 < \epsilon < 1$
in quasi-polynomial
$n^{O(\ln n - \ln \epsilon )}$
time if the Lipschitz constant of the non-linear part of f with respect to the
$\ell ^1$
metric on the Boolean cube does not exceed
$1-\delta $
, for any
$\delta>0$
, fixed in advance. For a cubic polynomial f, we get the same result under a somewhat stronger condition. We apply the method of polynomial interpolation, for which we prove that
$\sum _x e^{\tilde {f}(x)} \ne 0$
for complex-valued polynomials
$\tilde {f}$
in a neighborhood of a real-valued f satisfying the above mentioned conditions. The bounds are asymptotically optimal. Results on the zero-free region are interpreted as the absence of a phase transition in the Lee–Yang sense in the corresponding Ising model. The novel feature of the bounds is that they control the total interaction of each vertex but not every single interaction of sets of vertices.
We show that the cohomology intersection number of a twisted Gauss–Manin connection with regularization condition is a rational function. As an application, we obtain a new quadratic relation associated to period integrals of a certain family of K3 surfaces.
We present a polynomial-time Markov chain Monte Carlo algorithm for estimating the partition function of the antiferromagnetic Ising model on any line graph. The analysis of the algorithm exploits the ‘winding’ technology devised by McQuillan [CoRR abs/1301.2880 (2013)] and developed by Huang, Lu and Zhang [Proc. 27th Symp. on Disc. Algorithms (SODA16), 514–527]. We show that exact computation of the partition function is #P-hard, even for line graphs, indicating that an approximation algorithm is the best that can be expected. We also show that Glauber dynamics for the Ising model is rapidly mixing on line graphs, an example being the kagome lattice.
Drift analysis is one of the state-of-the-art techniques for the runtime analysis of randomized search heuristics (RSHs) such as evolutionary algorithms (EAs), simulated annealing, etc. The vast majority of existing drift theorems yield bounds on the expected value of the hitting time for a target state, for example the set of optimal solutions, without making additional statements on the distribution of this time. We address this lack by providing a general drift theorem that includes bounds on the upper and lower tail of the hitting time distribution. The new tail bounds are applied to prove very precise sharp-concentration results on the running time of a simple EA on standard benchmark problems, including the class of general linear functions. On all these problems, the probability of deviating by an r-factor in lower-order terms of the expected time decreases exponentially with r. The usefulness of the theorem outside the theory of RSHs is demonstrated by deriving tail bounds on the number of cycles in random permutations. All these results handle a position-dependent (variable) drift that was not covered by previous drift theorems with tail bounds. Finally, user-friendly specializations of the general drift theorem are given.
We give a fully polynomial-time randomized approximation scheme (FPRAS) for the number of bases in bicircular matroids. This is a natural class of matroids for which counting bases exactly is #P-hard and yet approximate counting can be done efficiently.
In this paper we propose a polynomial-time deterministic algorithm for approximately counting the k-colourings of the random graph G(n, d/n), for constant d>0. In particular, our algorithm computes in polynomial time a
$(1\pm n^{-\Omega(1)})$
-approximation of the so-called ‘free energy’ of the k-colourings of G(n, d/n), for
$k\geq (1+\varepsilon) d$
with probability
$1-o(1)$
over the graph instances.
Our algorithm uses spatial correlation decay to compute numerically estimates of marginals of the Gibbs distribution. Spatial correlation decay has been used in different counting schemes for deterministic counting. So far algorithms have exploited a certain kind of set-to-point correlation decay, e.g. the so-called Gibbs uniqueness. Here we deviate from this setting and exploit a point-to-point correlation decay. The spatial mixing requirement is that for a pair of vertices the correlation between their corresponding configurations becomes weaker with their distance.
Furthermore, our approach generalizes in that it allows us to compute the Gibbs marginals for small sets of nearby vertices. Also, we establish a connection between the fluctuations of the number of colourings of G(n, d/n) and the fluctuations of the number of short cycles and edges in the graph.
There has been substantial interest in estimating the value of a graph parameter, i.e. of a real-valued function defined on the set of finite graphs, by querying a randomly sampled substructure whose size is independent of the size of the input. Graph parameters that may be successfully estimated in this way are said to be testable or estimable, and the sample complexity qz = qz(ε) of an estimable parameter z is the size of a random sample of a graph G required to ensure that the value of z(G) may be estimated within an error of ε with probability at least 2/3. In this paper, for any fixed monotone graph property $\mathcal{P}= \text{Forb}\!(\mathcal{F}),$ we study the sample complexity of estimating a bounded graph parameter z that, for an input graph G, counts the number of spanning subgraphs of G that satisfy$\mathcal{P}$. To improve upon previous upper bounds on the sample complexity, we show that the vertex set of any graph that satisfies a monotone property $\mathcal{P}$ may be partitioned equitably into a constant number of classes in such a way that the cluster graph induced by the partition is not far from satisfying a natural weighted graph generalization of $\mathcal{P}$. Properties for which this holds are said to be recoverable, and the study of recoverable properties may be of independent interest.
For a rumour spreading protocol, the spread time is defined as the first time everyone learns the rumour. We compare the synchronous push&pull rumour spreading protocol with its asynchronous variant, and show that for any n-vertex graph and any starting vertex, the ratio between their expected spread times is bounded by $O({n^{1/3}}{\log ^{2/3}}n)$. This improves the $O(\sqrt n)$ upper bound of Giakkoupis, Nazari and Woelfel (2016). Our bound is tight up to a factor of O(log n), as illustrated by the string of diamonds graph. We also show that if, for a pair α, β of real numbers, there exist infinitely many graphs for which the two spread times are nα and nβ in expectation, then $0 \le \alpha \le 1$ and $\alpha \le \beta \le {1 \over 3} + {2 \over 3} \alpha $; and we show each such pair α, β is achievable.
Given complex numbers w1,…,wn, we define the weight w(X) of a set X of 0–1 vectors as the sum of $w_1^{x_1} \cdots w_n^{x_n}$ over all vectors (x1,…,xn) in X. We present an algorithm which, for a set X defined by a system of homogeneous linear equations with at most r variables per equation and at most c equations per variable, computes w(X) within relative error ∊ > 0 in (rc)O(lnn-ln∊) time provided $|w_j| \leq \beta (r \sqrt{c})^{-1}$ for an absolute constant β > 0 and all j = 1,…,n. A similar algorithm is constructed for computing the weight of a linear code over ${\mathbb F}_p$. Applications include counting weighted perfect matchings in hypergraphs, counting weighted graph homomorphisms, computing weight enumerators of linear codes with sparse code generating matrices, and computing the partition functions of the ferromagnetic Potts model at low temperatures and of the hard-core model at high fugacity on biregular bipartite graphs.
The Orthogonal Least Squares (OLS) algorithm is an efficient sparse recovery algorithm that has received much attention in recent years. On one hand, this paper considers that the OLS algorithm recovers the supports of sparse signals in the noisy case. We show that the OLS algorithm exactly recovers the support of $K$-sparse signal $\boldsymbol{x}$ from $\boldsymbol{y}=\boldsymbol{\unicode[STIX]{x1D6F7}}\boldsymbol{x}+\boldsymbol{e}$ in $K$ iterations, provided that the sensing matrix $\boldsymbol{\unicode[STIX]{x1D6F7}}$ satisfies the restricted isometry property (RIP) with restricted isometry constant (RIC) $\unicode[STIX]{x1D6FF}_{K+1}<1/\sqrt{K+1}$, and the minimum magnitude of the nonzero elements of $\boldsymbol{x}$ satisfies some constraint. On the other hand, this paper demonstrates that the OLS algorithm exactly recovers the support of the best $K$-term approximation of an almost sparse signal $\boldsymbol{x}$ in the general perturbations case, which means both $\boldsymbol{y}$ and $\boldsymbol{\unicode[STIX]{x1D6F7}}$ are perturbed. We show that the support of the best $K$-term approximation of $\boldsymbol{x}$ can be recovered under reasonable conditions based on the restricted isometry property (RIP).