IA Scholar Query: Perfect Matchings via Uniform Sampling in Regular Bipartite Graphs
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgSat, 13 Aug 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Hamiltonicity of random subgraphs of the hypercube
https://scholar.archive.org/work/ai5y2fqoonfjjlsnrdm5tl5i6a
We study Hamiltonicity in random subgraphs of the hypercube 𝒬^n. Our first main theorem is an optimal hitting time result. Consider the random process which includes the edges of 𝒬^n according to a uniformly chosen random ordering. Then, with high probability, as soon as the graph produced by this process has minimum degree 2k, it contains k edge-disjoint Hamilton cycles, for any fixed k∈ℕ. Secondly, we obtain a perturbation result: if H⊆𝒬^n satisfies δ(H)≥α n with α>0 fixed and we consider a random binomial subgraph 𝒬^n_p of 𝒬^n with p∈(0,1] fixed, then with high probability H∪𝒬^n_p contains k edge-disjoint Hamilton cycles, for any fixed k∈ℕ. In particular, both results resolve a long standing conjecture, posed e.g. by Bollobás, that the threshold probability for Hamiltonicity in the random binomial subgraph of the hypercube equals 1/2. Our techniques also show that, with high probability, for all fixed p∈(0,1] the graph 𝒬^n_p contains an almost spanning cycle. Our methods involve branching processes, the Rödl nibble, and absorption.Padraig Condon, Alberto Espuny Díaz, António Girão, Daniela Kühn, Deryk Osthuswork_ai5y2fqoonfjjlsnrdm5tl5i6aSat, 13 Aug 2022 00:00:00 GMTAnalyzing and Optimizing Shared Mobility Fleet Impacts
https://scholar.archive.org/work/lkzmoqsmqvas3hm2bg7dkwpek4
Passenger vehicles enable activity, but they generate unpriced negative externalities such as air emissions and traffic. Those externalities constitute a market failure that may justify policy intervention. Passenger vehicle travel, especially within urban areas, is being transformed by vehicle electrification and by shared mobility options offered by ridesourcing services such as Uber and Lyft. These transformations' impacts on externalities are unclear a priori, as is the role of policy to influence them. To investigate these externalities and what options can address them, I use a mixture of simulation and empirical analysis.Matthew Bruchonwork_lkzmoqsmqvas3hm2bg7dkwpek4Thu, 11 Aug 2022 00:00:00 GMTNearly Optimal Pseudorandomness From Hardness
https://scholar.archive.org/work/qiigm2fwhngjvam5agpfoa6sxq
Existing proofs that deduce BPP = P from circuit lower bounds convert randomized algorithms into deterministic algorithms with a large polynomial slowdown. We convert randomized algorithms into deterministic ones with little slowdown . Specifically, assuming exponential lower bounds against randomized NP ∩ coNP circuits, formally known as randomized SVN circuits, we convert any randomized algorithm over inputs of length n running in time t ≥ n into a deterministic one running in time t 2 + α for an arbitrarily small constant α > 0. Such a slowdown is nearly optimal for t close to n , since under standard complexity-theoretic assumptions, there are problems with an inherent quadratic derandomization slowdown. We also convert any randomized algorithm that errs rarely into a deterministic algorithm having a similar running time (with pre-processing). The latter derandomization result holds under weaker assumptions, of exponential lower bounds against deterministic SVN circuits. Our results follow from a new, nearly optimal, explicit pseudorandom generator fooling circuits of size s with seed length (1 + α )log s , under the assumption that there exists a function f ∈ E that requires randomized SVN circuits of size at least \(2^{(1-\alpha ^{\prime })n} \) , where α = O ( α ′). The construction uses, among other ideas, a new connection between pseudoentropy generators and locally list recoverable codes.Dean Doron, Dana Moshkovitz, Justin Oh, David Zuckermanwork_qiigm2fwhngjvam5agpfoa6sxqWed, 10 Aug 2022 00:00:00 GMTd-connectivity of the random graph with restricted budget
https://scholar.archive.org/work/dexqgflun5csfp5xbnn3wqmtlm
In this short note we consider a graph process recently introduced by Frieze, Krivelevich, and Michaeli. In their model, the edges of the complete graph K_n are ordered uniformly at random, and are then revealed consecutively to a player called Builder. At every round, Builder must decide if they accept the edge proposed at this round or not. We prove that for every d≥ 2 Builder may construct a spanning d-connected graph after (1+o(1))nlog n rounds by accepting at most (1+o(1))dn/2 edges with probability converging to 1 as n→∞. This settles a conjecture of Frieze, Krivelevich, and Michaeli.Lyuben Lichevwork_dexqgflun5csfp5xbnn3wqmtlmMon, 08 Aug 2022 00:00:00 GMTSubspace-Based Pilot Decontamination in User-Centric Scalable Cell-Free Wireless Networks
https://scholar.archive.org/work/ahnx24yky5hfplqci6pfd7quxe
We consider a cell-free wireless system operated in Time Division Duplex (TDD) mode with user-centric clusters of remote radio units (RUs). Since the uplink pilot dimensions per channel coherence slot is limited, co-pilot users might incur mutual pilot contamination. In the current literature, it is assumed that the long-term statistical knowledge of all user channels is available. This enables MMSE channel estimation or simplified dominant subspace projection, which achieves significant pilot decontamination under certain assumptions on the channel covariance matrices. However, estimating the channel covariance matrix or even just its dominant subspace at all RUs forming a user cluster is not an easy task. In fact, if not properly designed, a piloting scheme for such long-term statistics estimation will also be subject to the contamination problem. In this paper, we propose a new channel subspace estimation scheme explicitly designed for cell-free wireless networks. Our scheme is based on 1) a sounding reference signal (SRS) using latin squares wideband frequency hopping, and 2) a subspace estimation method based on robust Principal Component Analysis (R-PCA). The SRS hopping scheme ensures that for any user and any RU participating in its cluster, only a few pilot measurements will contain strong co-pilot interference. These few heavily contaminated measurements are (implicitly) eliminated by R-PCA, which is designed to regularize the estimation and discount the "outlier" measurements. Our simulation results show that the proposed scheme achieves almost perfect subspace knowledge, which in turns yields system performance very close to that with ideal channel state information, thus essentially solving the problem of pilot contamination in cell-free user-centric TDD wireless networks.Fabian Göttsch, Noboru Osawa, Takeo Ohseki, Kosuke Yamazaki, Giuseppe Cairework_ahnx24yky5hfplqci6pfd7quxeMon, 08 Aug 2022 00:00:00 GMTCombinatorial Optimization via the Sum of Squares Hierarchy
https://scholar.archive.org/work/ksux7wlwmndldojrnagqqkvcdu
We study the Sum of Squares (SoS) Hierarchy with a view towards combinatorial optimization. We survey the use of the SoS hierarchy to obtain approximation algorithms on graphs using their spectral properties. We present a simplified proof of the result of Feige and Krauthgamer on the performance of the hierarchy for the Maximum Clique problem on random graphs. We also present a result of Guruswami and Sinop that shows how to obtain approximation algorithms for the Minimum Bisection problem on low threshold-rank graphs. We study inapproximability results for the SoS hierarchy for general constraint satisfaction problems and problems involving graph densities such as the Densest k-subgraph problem. We improve the existing inapproximability results for general constraint satisfaction problems in the case of large arity, using stronger probabilistic analyses of expansion of random instances. We examine connections between constraint satisfaction problems and density problems on graphs. Using them, we obtain new inapproximability results for the hierarchy for the Densest k-subhypergraph problem and the Minimum p-Union problem, which are proven via reductions. We also illustrate the relatively new idea of pseudocalibration to construct integrality gaps for the SoS hierarchy for Maximum Clique and Max K-CSP. The application to Max K-CSP that we present is known in the community but has not been presented before in the literature, to the best of our knowledge.Goutham Rajendranwork_ksux7wlwmndldojrnagqqkvcduMon, 08 Aug 2022 00:00:00 GMTLozenge tilings and the Gaussian free field on a cylinder
https://scholar.archive.org/work/trhk4anxwzfd5eqfqo3laylr3a
We use the periodic Schur process, introduced in arXiv:math/0601019v1, to study the random height function of lozenge tilings (equivalently, dimers) on an infinite cylinder distributed under two variants of the q^vol measure. Under the first variant, corresponding to random cylindric partitions, the height function converges to a deterministic limit shape and fluctuations around it are given by the Gaussian free field in the conformal structure predicted by the Kenyon-Okounkov conjecture. Under the second variant, corresponding to an unrestricted dimer model on the cylinder, the fluctuations are given by the same Gaussian free field with an additional discrete Gaussian shift component. Fluctuations of the latter type have been previously conjectured for dimer models on planar domains with holes.Andrew Ahn, Marianna Russkikh, Roger Van Peskiwork_trhk4anxwzfd5eqfqo3laylr3aSat, 06 Aug 2022 00:00:00 GMTDescriptive vs. inferential community detection
https://scholar.archive.org/work/7kx6vshwkjabhnaukqodb5j6ka
Community detection is one of the most important methodological fields of network science, and one which has attracted a significant amount of attention over the past decades. This area deals with the automated division of a network into fundamental building blocks, with the objective of providing a summary of its large-scale structure. Despite its importance and widespread adoption, there is a noticeable gap between what is arguably the state-of-the-art and the methods that are actually used in practice in a variety of fields. Here we attempt to address this discrepancy by dividing existing methods according to whether they have a "descriptive" or an "inferential" goal. While descriptive methods find patterns in networks based on context-dependent notions of community structure, inferential methods articulate generative models, and attempt to fit them to data. In this way, they are able to provide insights into the mechanisms of network formation, and separate structure from randomness in a manner supported by statistical evidence. We review how employing descriptive methods with inferential aims is riddled with pitfalls and misleading answers, and thus should be in general avoided. We argue that inferential methods are more typically aligned with clearer scientific questions, yield more robust results, and should be in many cases preferred. We attempt to dispel some myths and half-truths often believed when community detection is employed in practice, in an effort to improve both the use of such methods as well as the interpretation of their results.Tiago P. Peixotowork_7kx6vshwkjabhnaukqodb5j6kaFri, 05 Aug 2022 00:00:00 GMTDirac-type theorems in random hypergraphs
https://scholar.archive.org/work/radpwtwh4replhtyszkmn4v6jm
For positive integers d0 and any "not too small" p, we prove that a random k-uniform hypergraph G with n vertices and edge probability p typically has the property that every spanning subgraph of G with minimum degree at least (1+ε)m_d(k,n)p has a perfect matching. One interesting aspect of our proof is a "non-constructive" application of the absorbing method, which allows us to prove a bound in terms of m_d(k,n) without actually knowing its value.Asaf Ferber, Matthew Kwanwork_radpwtwh4replhtyszkmn4v6jmThu, 04 Aug 2022 00:00:00 GMTSubstructures in Latin squares
https://scholar.archive.org/work/7s5oc53pq5g5djqancnbkzs7zm
We prove several results about substructures in Latin squares. First, we explain how to adapt our recent work on high-girth Steiner triple systems to the setting of Latin squares, resolving a conjecture of Linial that there exist Latin squares with arbitrarily high girth. As a consequence, we see that the number of order-n Latin squares with no intercalate (i.e., no 2×2 Latin subsquare) is at least (e^-9/4n-o(n))^n^2. Equivalently, ℙ[𝐍=0]≥ e^-n^2/4-o(n^2)=e^-(1+o(1))𝔼𝐍, where 𝐍 is the number of intercalates in a uniformly random order-n Latin square. In fact, extending recent work of Kwan, Sah, and Sawhney, we resolve the general large-deviation problem for intercalates in random Latin squares, up to constant factors in the exponent: for any constant 0<δ≤1 we have ℙ[𝐍≤(1-δ)𝔼𝐍]=exp(-Θ(n^2)) and for any constant δ>0 we have ℙ[𝐍≥(1+δ)𝔼𝐍]=exp(-Θ(n^4/3log n)). Finally, as an application of some new general tools for studying substructures in random Latin squares, we show that in almost all order-n Latin squares, the number of cuboctahedra (i.e., the number of pairs of possibly degenerate 2×2 submatrices with the same arrangement of symbols) is of order n^4, which is the minimum possible. As observed by Gowers and Long, this number can be interpreted as measuring "how associative" the quasigroup associated with the Latin square is.Matthew Kwan, Ashwin Sah, Mehtaab Sawhney, Michael Simkinwork_7s5oc53pq5g5djqancnbkzs7zmThu, 04 Aug 2022 00:00:00 GMTCentral Limit Theorem in Disordered Monomer-Dimer Model
https://scholar.archive.org/work/nraxfctomzdaboszssbh5rpjo4
We consider the disordered monomer-dimer model on general finite graphs with bounded degree, where both the edges and the vertices are equipped with i.i.d. random weights coming from two possibly different distributions. Under the finite fourth moment assumption on the weight distributions, we prove a Gaussian central limit theorem for the free energy of the associated Gibbs measure and also provide a rate of convergence in the Kolmogorov-Smirnov distance. The central limit theorem continues to hold under a nearly optimal finite (2+ϵ)-moment assumption on the weight distributions if the underlying graphs are further assumed to have a uniformly subexponential volume growth. This generalizes a recent result by Dey and Krishnan (arXiv:2109.12716) who showed a Gaussian central limit theorem in the disordered monomer-dimer model on cylinder graphs. Our proof relies on the idea that the disordered monomer-dimer model exhibits a decay of correlation with high probability.Wai-Kit Lam, Arnab Senwork_nraxfctomzdaboszssbh5rpjo4Wed, 03 Aug 2022 00:00:00 GMTQuantum Computing: Lecture Notes
https://scholar.archive.org/work/2pcfo6u7jzg25alp6mv6fq3w2y
This is a set of lecture notes suitable for a Master's course on quantum computation and information from the perspective of theoretical computer science. The first version was written in 2011, with many extensions and improvements in subsequent years. The first 10 chapters cover the circuit model and the main quantum algorithms (Deutsch-Jozsa, Simon, Shor, Hidden Subgroup Problem, Grover, quantum walks, Hamiltonian simulation and HHL). They are followed by 3 chapters about complexity, 4 chapters about distributed ("Alice and Bob") settings, a chapter about quantum machine learning, and a final chapter about quantum error correction. Appendices A and B give a brief introduction to the required linear algebra and some other mathematical and computer science background. All chapters come with exercises, with some hints provided in Appendix C.Ronald de Wolfwork_2pcfo6u7jzg25alp6mv6fq3w2yTue, 02 Aug 2022 00:00:00 GMTLet's Make Block Coordinate Descent Converge Faster: Faster Greedy Rules, Message-Passing, Active-Set Complexity, and Superlinear Convergence
https://scholar.archive.org/work/ioo5osphkzh6lkujg7daz2chhy
Block coordinate descent (BCD) methods are widely used for large-scale numerical optimization because of their cheap iteration costs, low memory requirements, amenability to parallelization, and ability to exploit problem structure. Three main algorithmic choices influence the performance of BCD methods: the block partitioning strategy, the block selection rule, and the block update rule. In this paper we explore all three of these building blocks and propose variations for each that can significantly improve the progress made by each BCD iteration. We (i) propose new greedy block-selection strategies that guarantee more progress per iteration than the Gauss-Southwell rule; (ii) explore practical issues like how to implement the new rules when using "variable" blocks; (iii) explore the use of message-passing to compute matrix or Newton updates efficiently on huge blocks for problems with sparse dependencies between variables; and (iv) consider optimal active manifold identification, which leads to bounds on the "active-set complexity" of BCD methods and leads to superlinear convergence for certain problems with sparse solutions (and in some cases finite termination at an optimal solution). We support all of our findings with numerical results for the classic machine learning problems of least squares, logistic regression, multi-class logistic regression, label propagation, and L1-regularization.Julie Nutini and Issam Laradji and Mark Schmidtwork_ioo5osphkzh6lkujg7daz2chhySun, 31 Jul 2022 00:00:00 GMTSharp thresholds for Ramsey properties
https://scholar.archive.org/work/ususdnwzwbgatcvw55m6wxi6jm
In this work, we develop a unified framework for establishing sharp threshold results for various Ramsey properties. To achieve this, we view such properties as non-colourability of auxiliary hypergraphs. Our main technical result gives sufficient conditions on a sequence of such hypergraphs that guarantee that this non-colourability property has a sharp threshold in subhypergraphs induced by random subsets of the vertices. Furthermore, we verify these conditions in several cases of interest. In the classical setting of Ramsey theory for graphs, we show that the property of being Ramsey for a graph H in r colours has a sharp threshold in G_n,p, for all r ≥ 2 and all H in a class of graphs that includes all cliques and cycles. In the arithmetic setting, we establish sharpness of thresholds for the properties corresponding to van der Waerden's theorem and Schur's theorem, also in any number of colours.Ehud Friedgut, Eden Kuperwasser, Wojciech Samotij, Mathias Schachtwork_ususdnwzwbgatcvw55m6wxi6jmThu, 28 Jul 2022 00:00:00 GMTColoring graphs with forbidden almost bipartite subgraphs
https://scholar.archive.org/work/ln7ahharsrfgjlgqehbz7kfuou
A conjecture of Alon, Krivelevich, and Sudakov states that, for any graph F, there is a constant c(F) > 0 such that χ(G) ≤ (c(F) + o(1)) Δ / logΔ for all F-free graphs G of maximum degree Δ. The only graphs F for which this conjecture has been verified so far – by Alon, Krivelevich, and Sudakov themselves – are the so-called almost bipartite graphs, i.e., graphs that can be made bipartite by removing at most one vertex. Equivalently, a graph is almost bipartite if it is a subgraph of the complete tripartite graph K_1,t,t for some t ∈ℕ. The best heretofore known upper bound on c(F) for almost bipartite F is due to Davies, Kang, Pirot, and Sereni, who showed that c(K_1,t,t) ≤ t. We prove that in fact c(F) ≤ 4 for any almost bipartite graph F, thus making the bound independent of F in all the known cases of the conjecture. We also establish a more general version of this result in the setting of DP-coloring (also known as correspondence coloring) and consider some algorithmic consequences.James Anderson, Anton Bernshteyn, Abhishek Dhawanwork_ln7ahharsrfgjlgqehbz7kfuouWed, 27 Jul 2022 00:00:00 GMTPerfect Matching in Random Graphs is as Hard as Tseitin
https://scholar.archive.org/work/yqt4a7mrgrfmbiddrqs25dgvlu
We study the complexity of proving that a sparse random regular graph on an odd number of vertices does not have a perfect matching, and related problems involving each vertex being matched some pre-specified number of times. We show that this requires proofs of degree Ω(n / log n) in the Polynomial Calculus (over fields of characteristic 2) and Sum-of-Squares proof systems, and exponential size in the bounded-depth Frege proof system. This resolves a question by Razborov asking whether the Lovász-Schrijver proof system requires n^δ rounds to refute these formulas for some δ > 0. The results are obtained by a worst-case to average-case reduction of these formulas relying on a topological embedding theorem which may be of independent interest.Per Austrin, Kilian Rissework_yqt4a7mrgrfmbiddrqs25dgvluWed, 27 Jul 2022 00:00:00 GMTNeural Design for Genetic Perturbation Experiments
https://scholar.archive.org/work/nxk4f4osgrfinhvakgtn7nxbey
The problem of how to genetically modify cells in order to maximize a certain cellular phenotype has taken center stage in drug development over the last few years (with, for example, genetically edited CAR-T, CAR-NK, and CAR-NKT cells entering cancer clinical trials). Exhausting the search space for all possible genetic edits (perturbations) or combinations thereof is infeasible due to cost and experimental limitations. This work provides a theoretically sound framework for iteratively exploring the space of perturbations in pooled batches in order to maximize a target phenotype under an experimental budget. Inspired by this application domain, we study the problem of batch query bandit optimization and introduce the Optimistic Arm Elimination (OAE) principle designed to find an almost optimal arm under different functional relationships between the queries (arms) and the outputs (rewards). We analyze the convergence properties of OAE by relating it to the Eluder dimension of the algorithm's function class and validate that OAE outperforms other strategies in finding optimal actions in experiments on simulated problems, public datasets well-studied in bandit contexts, and in genetic perturbation datasets when the regression model is a deep neural network. OAE also outperforms the benchmark algorithms in 3 of 4 datasets in the GeneDisco experimental planning challenge.Aldo Pacchiano, Drausin Wulsin, Robert A. Barton, Luis Volochwork_nxk4f4osgrfinhvakgtn7nxbeyTue, 26 Jul 2022 00:00:00 GMTDimers on Riemann surfaces I: Temperleyan forests
https://scholar.archive.org/work/qby3cdrubbfallpfq7d4hfda3a
This is the first article in a series of two papers in which we study the Temperleyan dimer model on an arbitrary bounded Riemann surface of finite topolgical type. The end goal of both papers is to prove the convergence of height fluctuations to a universal and conformally invariant scaling limit. In this part we show that the dimer model on the Temperleyan superposition of a graph embedded on the surface and its dual is well posed, provided that we remove an appropriate number of punctures. We further show that the resulting dimer configuration is in bijection with an object which we call Temperleyan forest, whose law is characterised in terms of a certain topological condition. Finally we discuss the relation between height differences and Temperleyan forest, and give a criterion guaranteeing the convergence of the height fluctuations in terms of the Temperleyan forest.Nathanaël Berestycki, Benoit Laslier, Gourab Raywork_qby3cdrubbfallpfq7d4hfda3aWed, 20 Jul 2022 00:00:00 GMTImproved mixing for the convex polygon triangulation flip walk
https://scholar.archive.org/work/7ekbyaj7ujgeveh6sqd3ok4jyy
We prove that the well-studied triangulation flip walk on a convex point set mixes in time O(n^4.75), the first progress since McShine and Tetali's O(n^5 log n) bound in 1997. In the process we determine the expansion of the associahedron graph K_n up to a factor of O(n^3/4). To obtain these results, we extend a framework we developed in a previous preprint–extending the projection-restriction technique of Jerrum, Son, Tetali, and Vigoda–for establishing conditions under which the Glauber dynamics on independent sets and other combinatorial structures mix rapidly.David Eppstein, Daniel Frishbergwork_7ekbyaj7ujgeveh6sqd3ok4jyyWed, 20 Jul 2022 00:00:00 GMTSublinear Algorithms and Lower Bounds for Estimating MST and TSP Cost in General Metrics
https://scholar.archive.org/work/3rs6yhfilrbl7mv7ho4cbzj4ee
We consider the design of sublinear space and query complexity algorithms for estimating the cost of a minimum spanning tree (MST) and the cost of a minimum traveling salesman (TSP) tour in a metric on n points. We first consider the o(n)-space regime and show that, when the input is a stream of all n2 entries of the metric, for any α≥ 2, both MST and TSP cost can be α-approximated using Õ(n/α) space, and that Ω(n/α^2) space is necessary for this task. Moreover, we show that even if the streaming algorithm is allowed p passes over a metric stream, it still requires Ω̃(√(n/α p^2)) space. We next consider the semi-streaming regime, where computing even the exact MST cost is easy and the main challenge is to estimate TSP cost to within a factor that is strictly better than 2. We show that, if the input is a stream of all edges of the weighted graph that induces the underlying metric, for any ε > 0, any one-pass (2-ε)-approximation of TSP cost requires Ω(ε^2 n^2) space; on the other hand, there is an Õ(n) space two-pass algorithm that approximates the TSP cost to within a factor of 1.96. Finally, we consider the query complexity of estimating metric TSP cost to within a factor that is strictly better than 2, when the algorithm is given access to a matrix that specifies pairwise distances between all points. For MST estimation in this model, it is known that a (1+ε)-approximation is achievable with Õ(n/ε^O(1)) queries. We design an algorithm that performs Õ(n^1.5) distance queries and achieves a strictly better than 2-approximation when either the metric is known to contain a spanning tree supported on weight-1 edges or the algorithm is given access to a minimum spanning tree of the graph.Yu Chen, Sanjeev Khanna, Zihan Tanwork_3rs6yhfilrbl7mv7ho4cbzj4eeWed, 20 Jul 2022 00:00:00 GMT