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Beating 1-1/e for ordered prophets 击败1-1/e的有序先知
Pub Date : 2017-04-19 DOI: 10.1145/3055399.3055479
M. Abolhassani, S. Ehsani, Hossein Esfandiari, M. Hajiaghayi, Robert D. Kleinberg, Brendan Lucier
Hill and Kertz studied the prophet inequality on iid distributions [The Annals of Probability 1982]. They proved a theoretical bound of 1 - 1/e on the approximation factor of their algorithm. They conjectured that the best approximation factor for arbitrarily large n is 1/1+1/e ≃ 0.731. This conjecture remained open prior to this paper for over 30 years. In this paper we present a threshold-based algorithm for the prophet inequality with n iid distributions. Using a nontrivial and novel approach we show that our algorithm is a 0.738-approximation algorithm. By beating the bound of 1/1+1/e, this refutes the conjecture of Hill and Kertz. Moreover, we generalize our results to non-uniform distributions and discuss its applications in mechanism design.
Hill和Kertz研究了iid分布上的先知不等式[the Annals of Probability 1982]。他们证明了他们的算法的近似因子的理论边界为1 - 1/e。他们推测任意大n的最佳近似因子是1/1+1/e≃0.731。在这篇论文发表之前,这个猜想已经存在了30多年。本文提出了一种基于阈值的n - id分布的预测不等式算法。使用一种非平凡的新颖方法,我们证明了我们的算法是0.738近似算法。通过突破1/1+1/e的界限,反驳了Hill和Kertz的猜想。此外,我们将结果推广到非均匀分布,并讨论了其在机构设计中的应用。
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引用次数: 76
Stability of service under time-of-use pricing 分时定价下的服务稳定性
Pub Date : 2017-04-07 DOI: 10.1145/3055399.3055455
Shuchi Chawla, Nikhil R. Devanur, A. Holroyd, Anna R. Karlin, James B. Martin, Balasubramanian Sivan
We consider time-of-use pricing as a technique for matching supply and demand of temporal resources with the goal of maximizing social welfare. Relevant examples include energy, computing resources on a cloud computing platform, and charging stations for electric vehicles, among many others. A client/job in this setting has a window of time during which he needs service, and a particular value for obtaining it. We assume a stochastic model for demand, where each job materializes with some probability via an independent Bernoulli trial. Given a per-time-unit pricing of resources, any realized job will first try to get served by the cheapest available resource in its window and, failing that, will try to find service at the next cheapest available resource, and so on. Thus, the natural stochastic fluctuations in demand have the potential to lead to cascading overload events. Our main result shows that setting prices so as to optimally handle the expected demand works well: with high probability, when the actual demand is instantiated, the system is stable and the expected value of the jobs served is very close to that of the optimal offline algorithm.
我们认为使用时间定价是一种以社会福利最大化为目标来匹配时间资源供需的技术。相关的例子包括能源、云计算平台上的计算资源、电动汽车充电站等。在此设置中,客户机/作业有一个需要服务的时间窗口,以及获取服务的特定值。我们假设需求是一个随机模型,通过独立的伯努利试验,每个工作都有一定的概率实现。给定资源的每时间单位定价,任何已实现的作业将首先尝试由其窗口中最便宜的可用资源提供服务,如果失败,将尝试在下一个最便宜的可用资源处寻找服务,依此类推。因此,需求的自然随机波动有可能导致级联过载事件。我们的主要结果表明,设定价格以最优地处理预期需求效果很好:当实际需求实例化时,系统很有可能是稳定的,所服务的工作的期望值与最优离线算法非常接近。
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引用次数: 28
The integrality gap of the Goemans-Linial SDP relaxation for sparsest cut is at least a constant multiple of √log n 对于最稀疏切割,Goemans-Linial SDP松弛的完整性间隙至少是√log n的常数倍
Pub Date : 2017-04-04 DOI: 10.1145/3055399.3055413
A. Naor, Robert Young
We prove that the integrality gap of the Goemans-Linial semidefinite programming relaxation for the Sparsest Cut Problem is Ω(√logn) on inputs with n vertices, thus matching the previously best known upper bound (logn)1/2+o(1) up to lower-order factors. This statement is a consequence of the following new isoperimetric-type inequality. Consider the 8-regular graph whose vertex set is the 5-dimensional integer grid ℤ5 and where each vertex (a,b,c,d,e)∈ ℤ5 is connected to the 8 vertices (a± 1,b,c,d,e), (a,b± 1,c,d,e), (a,b,c± 1,d,e± a), (a,b,c,d± 1,e± b). This graph is known as the Cayley graph of the 5-dimensional discrete Heisenberg group. Given Ω⊆ ℤ5, denote the size of its edge boundary in this graph (a.k.a. the horizontal perimeter of Ω) by |∂hΩ|. For t ϵ ℕ, denote by |∂vtΩ| the number of (a,b,c,d,e)ϵ ℤ5 such that exactly one of the two vectors (a,b,c,d,e),(a,b,c,d,e+t) is in Ω. The vertical perimeter of Ω is defined to be |∂vΩ|= √Σt=1∞|∂vtΩ|2/t2. We show that every subset Ω⊆ ℤ5 satisfies |∂vΩ|=O(|∂hΩ|). This vertical-versus-horizontal isoperimetric inequality yields the above-stated integrality gap for Sparsest Cut and answers several geometric and analytic questions of independent interest. The theorem stated above is the culmination of a program whose aim is to understand the performance of the Goemans-Linial semidefinite program through the embeddability properties of Heisenberg groups. These investigations have mathematical significance even beyond their established relevance to approximation algorithms and combinatorial optimization. In particular they contribute to a range of mathematical disciplines including functional analysis, geometric group theory, harmonic analysis, sub-Riemannian geometry, geometric measure theory, ergodic theory, group representations, and metric differentiation. This article builds on the above cited works, with the "twist" that while those works were equally valid for any finite dimensional Heisenberg group, our result holds for the Heisenberg group of dimension 5 (or higher) but fails for the 3-dimensional Heisenberg group. This insight leads to our core contribution, which is a deduction of an endpoint L1-boundedness of a certain singular integral on ℝ5 from the (local) L2-boundedness of the corresponding singular integral on ℝ3. To do this, we devise a corona-type decomposition of subsets of a Heisenberg group, in the spirit of the construction that David and Semmes performed in ℝn, but with two main conceptual differences (in addition to more technical differences that arise from the peculiarities of the geometry of Heisenberg group). Firstly, the "atoms" of our decomposition are perturbations of intrinsic Lipschitz graphs in the sense of Franchi, Serapioni, and Serra Cassano (plus the requisite "wild" regions that satisfy a Carleson packing condition). Secondly, we control the local overlap of our corona decomposition by using quantitative monotonicity rather than Jones-type β-numbers.
我们证明了最稀疏切割问题的Goemans-Linial半定规划松弛在有n个顶点的输入上的完整性缺口为Ω(√logn),从而匹配了之前已知的上界(logn)1/2+o(1)直到低阶因子。这个表述是下列新的等周型不等式的结果。考虑一个8正则图,它的顶点集是5维整数网格,其中每个顶点(a,b,c,d,e)∈t5与8个顶点(a±1,b,c,d,e), (a,b±1,c,d,e), (a,b,c, c±1,e±a), (a,b,c,d±1,e±b)相连。这个图被称为5维离散Heisenberg群的Cayley图。给定Ω,其边缘边界的大小(即Ω的水平周长)用|∂hΩ|表示。对于t λ∈,用|∂vtΩ|表示(a,b,c,d,e)的个数,使得两个向量(a,b,c,d,e),(a,b,c,d,e+t)恰好有一个在Ω中。Ω的垂直周长定义为|∂vΩ|=√Σt=1∞|∂vtΩ|2/t2。我们证明了每个子集Ω都满足|∂vΩ|=O(|∂hΩ|)。这个垂直与水平等周不等式产生了上述的稀疏切割的完整性差距,并回答了几个独立感兴趣的几何和分析问题。上述定理是一个程序的高潮,其目的是通过海森堡群的可嵌入性来理解Goemans-Linial半定程序的性能。这些研究具有数学意义,甚至超越了它们与近似算法和组合优化的既定相关性。特别是,他们对一系列数学学科做出了贡献,包括泛函分析、几何群论、调和分析、亚黎曼几何、几何测度理论、遍历理论、群表示和度量微分。这篇文章建立在上述引用的作品的基础上,“扭曲”的是,虽然这些作品对任何有限维的海森堡群都同样有效,但我们的结果适用于5维(或更高)的海森堡群,但不适用于3维的海森堡群。这一洞见引出了我们的核心贡献,即从相应的(局部)l2有界性推导出了某一个在∈5上的奇异积分的端点l1有界性。为了做到这一点,我们设计了一个海森堡群子集的冕型分解,本着David和Semmes在h - n中进行的构造的精神,但有两个主要的概念差异(除了海森堡群的几何特性引起的更多技术差异)。首先,我们分解的“原子”是Franchi, Serapioni和Serra Cassano意义上的本征Lipschitz图的扰动(加上满足Carleson填充条件的必要“野”区域)。其次,我们利用定量单调性而不是琼斯型β数来控制电晕分解的局部重叠。
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引用次数: 9
Synchronization strings: codes for insertions and deletions approaching the Singleton bound 同步字符串:接近单例绑定的插入和删除代码
Pub Date : 2017-04-03 DOI: 10.1145/3055399.3055498
Bernhard Haeupler, Amirbehshad Shahrasbi
We introduce synchronization strings, which provide a novel way of efficiently dealing with synchronization errors, i.e., insertions and deletions. Synchronization errors are strictly more general and much harder to deal with than more commonly considered half-errors, i.e., symbol corruptions and erasures. For every ε > 0, synchronization strings allow to index a sequence with an ε-O(1) size alphabet such that one can efficiently transform k synchronization errors into (1 + ε)k half-errors. This powerful new technique has many applications. In this paper we focus on designing insdel codes, i.e., error correcting block codes (ECCs) for insertion deletion channels. While ECCs for both half-errors and synchronization errors have been intensely studied, the later has largely resisted progress. As Mitzenmacher puts it in his 2009 survey: "Channels with synchronization errors ... are simply not adequately understood by current theory. Given the near-complete knowledge we have for channels with erasures and errors ... our lack of understanding about channels with synchronization errors is truly remarkable." Indeed, it took until 1999 for the first insdel codes with constant rate, constant distance, and constant alphabet size to be constructed and only since 2016 are there constructions of constant rate indel codes for asymptotically large noise rates. Even in the asymptotically large or small noise regime these codes are polynomially far from the optimal rate-distance tradeoff. This makes the understanding of insdel codes up to this work equivalent to what was known for regular ECCs after Forney introduced concatenated codes in his doctoral thesis 50 years ago. A straight forward application of our synchronization strings based indexing method gives a simple black-box construction which transforms any ECC into an equally efficient insdel code with only a small increase in the alphabet size. This instantly transfers much of the highly developed understanding for regular ECCs over large constant alphabets into the realm of insdel codes. Most notably, for the complete noise spectrum we obtain efficient "near-MDS" insdel codes which get arbitrarily close to the optimal rate-distance tradeoff given by the Singleton bound. In particular, for any δ ∈ (0,1) and ε > 0 we give insdel codes achieving a rate of 1 - ξ - ε over a constant size alphabet that efficiently correct a δ fraction of insertions or deletions.
我们引入了同步字符串,它提供了一种有效处理同步错误的新方法,即插入和删除。同步错误严格来说比一般认为的半错误(即符号损坏和擦除)更通用,也更难处理。对于每个ε > 0,同步字符串允许用ε- o(1)大小的字母表索引序列,这样可以有效地将k个同步误差转换为(1 + ε)k个半误差。这项强大的新技术有许多用途。在本文中,我们专注于设计插入码,即插入删除信道的纠错块码(ECCs)。虽然半误差和同步误差的ECCs已经得到了广泛的研究,但后者在很大程度上阻碍了进展。正如米岑马赫在2009年的调查中所说:“有同步错误的频道……是目前的理论无法充分理解的。鉴于我们对带有擦除和错误的通道的近乎完整的了解……我们对存在同步错误的通道缺乏了解,这一点确实值得注意。”实际上,直到1999年才构造出第一个具有恒定速率、恒定距离和恒定字母大小的indel码,直到2016年才构造出具有渐近大噪声率的恒定速率indel码。即使在渐近的大噪声或小噪声条件下,这些码也多项式地远离最佳速率-距离权衡。这使得对这项工作的内部代码的理解相当于50年前Forney在他的博士论文中引入串联代码后对常规ecc的理解。我们基于同步字符串的索引方法的直接应用程序提供了一个简单的黑盒结构,它将任何ECC转换为同样高效的内部代码,仅增加了少量的字母大小。这立即将对大型常量字母的常规ecc的高度理解转移到内部代码领域。最值得注意的是,对于完整的噪声谱,我们获得了有效的“近mds”内码,它可以任意接近由单例界给出的最佳速率-距离权衡。特别地,对于任意δ∈(0,1)和ε >,我们给出了在恒定大小的字母表上实现1 - ξ - ε率的插入码,有效地纠正了δ分数的插入或删除。
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引用次数: 67
Complexity of short Presburger arithmetic 短Presburger算法的复杂度
Pub Date : 2017-04-02 DOI: 10.1145/3055399.3055435
Danny Nguyen, I. Pak
We study complexity of short sentences in Presburger arithmetic (Short-PA). Here by “short” we mean sentences with a bounded number of variables, quantifers, inequalities and Boolean operations; the input consists only of the integers involved in the inequalities. We prove that assuming Kannan’s partition can be found in polynomial time, the satisfability of Short-PA sentences can be decided in polynomial time. Furthermore, under the same assumption, we show that the numbers of satisfying assignments of short Presburger sentences can also be computed in polynomial time.
本文研究了Presburger算法(short - pa)中短句的复杂性。这里所说的“短”是指具有有限数量的变量、量词、不等式和布尔运算的句子;输入仅由不等式中涉及的整数组成。我们证明了假设在多项式时间内可以找到Kannan划分,则可以在多项式时间内确定短句的可满足性。此外,在相同的假设下,我们证明了短Presburger句的满足赋值的数量也可以在多项式时间内计算出来。
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引用次数: 11
Finding even cycles faster via capped k-walks 通过限定k步法更快地找到偶数周期
Pub Date : 2017-03-30 DOI: 10.1145/3055399.3055459
Søren Dahlgaard, M. B. T. Knudsen, Morten Stöckel
Finding cycles in graphs is a fundamental problem in algorithmic graph theory. In this paper, we consider the problem of finding and reporting a cycle of length 2k in an undirected graph G with n nodes and m edges for constant k≥ 2. A classic result by Bondy and Simonovits [J. Combinatorial Theory, 1974] implies that if m ≥ 100k n1+1/k, then G contains a 2k-cycle, further implying that one needs to consider only graphs with m = O(n1+1/k). Previously the best known algorithms were an O(n2) algorithm due to Yuster and Zwick [J. Discrete Math 1997] as well as a O(m2-(1+⌈ k/2 ⌉-1)/(k+1)) algorithm by Alon et. al. [Algorithmica 1997]. We present an algorithm that uses O( m2k/(k+1) ) time and finds a 2k-cycle if one exists. This bound is O(n2) exactly when m = Θ(n1+1/k). When finding 4-cycles our new bound coincides with Alon et. al., while for every k>2 our new bound yields a polynomial improvement in m. Yuster and Zwick noted that it is "plausible to conjecture that O(n2) is the best possible bound in terms of n". We show "conditional optimality": if this hypothesis holds then our O(m2k/(k+1)) algorithm is tight as well. Furthermore, a folklore reduction implies that no combinatorial algorithm can determine if a graph contains a 6-cycle in time O(m3/2-ε) for any ε>0 unless boolean matrix multiplication can be solved combinatorially in time O(n3-ε′) for some ε′ > 0, which is widely believed to be false. Coupled with our main result, this gives tight bounds for finding 6-cycles combinatorially and also separates the complexity of finding 4- and 6-cycles giving evidence that the exponent of m in the running time should indeed increase with k. The key ingredient in our algorithm is a new notion of capped k-walks, which are walks of length k that visit only nodes according to a fixed ordering. Our main technical contribution is an involved analysis proving several properties of such walks which may be of independent interest.
图中求环是算法图论中的一个基本问题。本文考虑在一个有n个节点和m条边且k≥2的无向图G中寻找和报告一个长度为2k的循环的问题。Bondy和Simonovits的一个经典结果[J]。组合理论,1974]表明,如果m≥100k n1+1/k,则G包含一个2k周期,进一步表明只需考虑m = O(n1+1/k)的图。以前最著名的算法是由Yuster和Zwick提出的O(n2)算法[J]。离散数学1997]以及由Alon等人[Algorithmica 1997]编写的O(m2-(1+ (k /2²)-1)/(k+1))算法。我们提出了一个算法,该算法使用O(m2k/(k+1))时间并找到一个2k周期(如果存在)。当m = Θ(n1+1/k)时,这个边界是O(n2)当发现4个循环时,我们的新边界与Alon等人一致,而对于每一个k>2,我们的新边界在m上产生一个多项式改进。Yuster和Zwick指出,“推测O(n2)是关于n的最佳可能边界是合理的”。我们展示了“条件最优性”:如果这个假设成立,那么我们的O(m2k/(k+1))算法也是严密的。此外,一个民间约简表明,没有组合算法可以确定一个图是否在时间O(m3/2-ε)上包含一个6周期,对于任何ε>0,除非布尔矩阵乘法可以在时间O(n3-ε ')上对某些ε ' >0进行组合求解,这被广泛认为是错误的。结合我们的主要结果,这给出了寻找6个周期组合的严格界限,也分离了寻找4和6个周期的复杂性,这证明了运行时间中m的指数确实应该随着k的增加而增加。我们算法的关键成分是一个新的概念,即长度为k的行走,根据固定的顺序只访问节点。我们的主要技术贡献是一个复杂的分析,证明了这种行走的几个特性,这些特性可能是独立的兴趣。
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引用次数: 20
Distributed exact shortest paths in sublinear time 亚线性时间内的精确分布最短路径
Pub Date : 2017-03-06 DOI: 10.1145/3055399.3055452
Michael Elkin
The distributed single-source shortest paths problem is one of the most fundamental and central problems in the message-passing distributed computing. Classical Bellman-Ford algorithm solves it in O(n) time, where n is the number of vertices in the input graph G. Peleg and Rubinovich, FOCS'99, showed a lower bound of Ω(D + √n) for this problem, where D is the hop-diameter of G. Whether or not this problem can be solved in o(n) time when D is relatively small is a major notorious open question. Despite intensive research that yielded near-optimal algorithms for the approximate variant of this problem, no progress was reported for the original problem. In this paper we answer this question in the affirmative. We devise an algorithm that requires O((n logn)5/6) time, for D = O(√n logn), and O(D1/3 #183; (n logn)2/3) time, for larger D. This running time is sublinear in n in almost the entire range of parameters, specifically, for D = o(n/log2 n). We also generalize our result in two directions. One is when edges have bandwidth b ≥ 1, and the other is the s-sources shortest paths problem. For the former problem, our algorithm provides an improved bound, compared to the unit-bandwidth case. In particular, we provide an all-pairs shortest paths algorithm that requires O(n5/3 #183; log2/3 n) time, even for b = 1, for all values of D. For the latter problem (of s sources), our algorithm also provides bounds that improve upon the previous state-of-the-art in the entire range of parameters. From the technical viewpoint, our algorithm computes a hopset G″ of a skeleton graph G′ of G without first computing G′ itself. We then conduct a Bellman-Ford exploration in G′ ∪ G″, while computing the required edges of G′ on the fly. As a result, our algorithm computes exactly those edges of G′ that it really needs, rather than computing approximately the entire G′.
分布式单源最短路径问题是消息传递分布式计算中最基本、最核心的问题之一。经典Bellman-Ford算法在O(n)时间内求解该问题,其中n为输入图g中的顶点数。Peleg和Rubinovich, FOCS'99给出了该问题的下界Ω(D +√n),其中D为g的跳直径。当D相对较小时,该问题能否在O(n)时间内求解是一个著名的开放问题。尽管对这个问题的近似变体进行了深入的研究,得出了接近最优的算法,但对于原始问题却没有任何进展。本文对这个问题作了肯定的回答。我们设计了一个需要O((n logn)5/6)时间的算法,对于D = O(√n logn)和O(D1/3 #183;(n logn)2/3)时间,对于更大的D,这个运行时间在n中几乎在整个参数范围内都是次线性的,特别是对于D = o(n/ log2n)。我们还在两个方向上推广我们的结果。一类是边带宽b≥1,另一类是s源最短路径问题。对于前一个问题,与单位带宽情况相比,我们的算法提供了一个改进的边界。特别地,我们提供了一个全对最短路径算法,该算法需要O(n5/3 #183;对于d的所有值,即使b = 1,也需要log2/3 (n)的时间。对于后一个问题(s个源),我们的算法还提供了在整个参数范围内改进之前的最新技术的边界。从技术角度来看,我们的算法在不首先计算G '本身的情况下,计算G的骨架图G '的hopset G″。然后我们在G '∪G″中进行Bellman-Ford探索,同时动态计算G '的所需边。因此,我们的算法精确地计算它真正需要的G '的那些边,而不是近似地计算整个G '。
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引用次数: 74
An SDP-based algorithm for linear-sized spectral sparsification 一种基于sdp的线性大小光谱稀疏化算法
Pub Date : 2017-02-27 DOI: 10.1145/3055399.3055477
Y. Lee, He Sun
For any undirected and weighted graph G=(V,E,w) with n vertices and m edges, we call a sparse subgraph H of G, with proper reweighting of the edges, a (1+ε)-spectral sparsifier if (1-ε)xTLGx≤xT LH x≤(1+ε)xTLGx holds for any xΕℝn, where LG and LH are the respective Laplacian matrices of G and H. Noticing that Ω(m) time is needed for any algorithm to construct a spectral sparsifier and a spectral sparsifier of G requires Ω(n) edges, a natural question is to investigate, for any constant ε, if a (1+ε)-spectral sparsifier of G with O(n) edges can be constructed in Ο(m) time, where the Ο notation suppresses polylogarithmic factors. All previous constructions on spectral sparsification require either super-linear number of edges or m1+Ω(1) time. In this work we answer this question affirmatively by presenting an algorithm that, for any undirected graph G and ε>0, outputs a (1+ε)-spectral sparsifier of G with O(n/ε2) edges in Ο(m/εO(1)) time. Our algorithm is based on three novel techniques: (1) a new potential function which is much easier to compute yet has similar guarantees as the potential functions used in previous references; (2) an efficient reduction from a two-sided spectral sparsifier to a one-sided spectral sparsifier; (3) constructing a one-sided spectral sparsifier by a semi-definite program.
任何无向加权图G = (V, E, w)与n顶点和m边,我们称之为稀疏H G的子图,通过适当的边的权重,一个(我µ1 +)光谱sparsifier如果(1 -ε)xTLGx≤xT LH x≤(1 +ε)xTLGx适用于任何xΕℝn, LG和LH各自的拉普拉斯算子矩阵G和H .注意到Ω(m)所需时间是任何算法来构造一个光谱sparsifier和光谱sparsifier G需要Ω(n)的边缘,一个自然的问题是调查,对于任何常数ε,如果可以在Ο(m)时间内构造具有O(n)条边的G的(1+ε)-谱稀疏子,其中Ο符号抑制了多对数因子。所有先前的光谱稀疏化结构都需要超线性边缘数或m1+Ω(1)时间。本文给出了一种算法,该算法对任意无向图G且ε>0,在Ο(m/εO(1))时间内输出具有O(n/ε2)条边的G的(1+ε)-谱稀疏子。我们的算法基于三种新技术:(1)一个新的势函数,它更容易计算,并且具有与以前参考文献中使用的势函数相似的保证;(2)将双面光谱稀疏器有效地简化为单面光谱稀疏器;(3)用半确定程序构造单侧谱稀疏器。
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引用次数: 76
Beyond Talagrand functions: new lower bounds for testing monotonicity and unateness 超越talagand函数:检验单调性和单调性的新下界
Pub Date : 2017-02-22 DOI: 10.1145/3055399.3055461
Xi Chen, Erik Waingarten, Jinyu Xie
We prove a lower bound of Ω(n1/3) for the query complexity of any two-sided and adaptive algorithm that tests whether an unknown Boolean function f:{0,1}n→ {0,1} is monotone versus far from monotone. This improves the recent lower bound of Ω(n1/4) for the same problem by Belovs and Blais (STOC'16). Our result builds on a new family of random Boolean functions that can be viewed as a two-level extension of Talagrand's random DNFs. Beyond monotonicity we prove a lower bound of Ω(√n) for two-sided, adaptive algorithms and a lower bound of Ω(n) for one-sided, non-adaptive algorithms for testing unateness, a natural generalization of monotonicity. The latter matches the linear upper bounds by Khot and Shinkar (RANDOM'16) and by Baleshzar, Chakrabarty, Pallavoor, Raskhodnikova, and Seshadhri (2017).
我们证明了任意检验未知布尔函数f:{0,1}n→{0,1}是否为单调与远非单调的双边自适应算法查询复杂度的下界Ω(n1/3)。这改进了最近Belovs和Blais (STOC'16)对相同问题的Ω(n /4)的下界。我们的结果建立在一组新的随机布尔函数的基础上,这些随机布尔函数可以看作是Talagrand随机dnf的两级扩展。除了单调性之外,我们证明了双面自适应算法的下界Ω(√n)和单面非自适应算法的下界Ω(n),用于测试单调性,这是单调性的自然推广。后者与Khot和Shinkar (RANDOM'16)以及Baleshzar、Chakrabarty、Pallavoor、Raskhodnikova和Seshadhri(2017)的线性上界相匹配。
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引用次数: 42
A generalization of permanent inequalities and applications in counting and optimization 永久不等式的推广及其在计数和优化中的应用
Pub Date : 2017-02-09 DOI: 10.1145/3055399.3055469
Nima Anari, S. Gharan
A polynomial pΕℝ[z1,…,zn] is real stable if it has no roots in the upper-half complex plane. Gurvits's permanent inequality gives a lower bound on the coefficient of the z1z2…zn monomial of a real stable polynomial p with nonnegative coefficients. This fundamental inequality has been used to attack several counting and optimization problems. Here, we study a more general question: Given a stable multilinear polynomial p with nonnegative coefficients and a set of monomials S, we show that if the polynomial obtained by summing up all monomials in S is real stable, then we can lower bound the sum of coefficients of monomials of p that are in S. We also prove generalizations of this theorem to (real stable) polynomials that are not multilinear. We use our theorem to give a new proof of Schrijver's inequality on the number of perfect matchings of a regular bipartite graph, generalize a recent result of Nikolov and Singh, and give deterministic polynomial time approximation algorithms for several counting problems.
一个多项式pΕ∈[z1,…,zn]是实稳定的,如果它在复平面的上半部分没有根。Gurvits永久不等式给出了非负系数实稳定多项式p的z1z2…zn单项式的系数下界。这个基本不等式已经被用来解决几个计数和优化问题。在这里,我们研究了一个更一般的问题:给定一个非负系数的稳定的多元线性多项式p和一组单项式S,我们证明了如果将S中所有单项式相加得到的多项式是实稳定的,那么我们可以给S中p的单项式的系数和下界。我们也证明了这个定理推广到非多元的(实稳定的)多项式。利用该定理给出了关于正则二部图完美匹配数的Schrijver不等式的一个新的证明,推广了Nikolov和Singh的最新结果,并给出了若干计数问题的确定性多项式时间逼近算法。
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引用次数: 58
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Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing
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