Pub Date : 2023-03-21DOI: https://dl.acm.org/doi/10.1145/3580475
Haitao Wang
Given a set of pairwise disjoint polygonal obstacles in the plane, finding an obstacle-avoiding Euclidean shortest path between two points is a classical problem in computational geometry and has been studied extensively. Previously, Hershberger and Suri (in SIAM Journal on Computing, 1999) gave an algorithm of O(n log n) time and O(n log n) space, where n is the total number of vertices of all obstacles. Recently, by modifying Hershberger and Suri’s algorithm, Wang (in SODA’21) reduced the space to O(n) while the runtime of the algorithm is still O(n log n). In this article, we present a new algorithm of O(n+h log h) time and O(n) space, provided that a triangulation of the free space is given, where h is the number of obstacles. The algorithm is better than the previous work when h is relatively small. Our algorithm builds a shortest path map for a source point s so that given any query point t, the shortest path length from s to t can be computed in O(log n) time and a shortest s-t path can be produced in additional time linear in the number of edges of the path.
{"title":"A New Algorithm for Euclidean Shortest Paths in the Plane","authors":"Haitao Wang","doi":"https://dl.acm.org/doi/10.1145/3580475","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3580475","url":null,"abstract":"<p>Given a set of pairwise disjoint polygonal obstacles in the plane, finding an obstacle-avoiding Euclidean shortest path between two points is a classical problem in computational geometry and has been studied extensively. Previously, Hershberger and Suri (in <i>SIAM Journal on Computing</i>, 1999) gave an algorithm of <i>O(n</i> log <i>n</i>) time and <i>O(n</i> log <i>n</i>) space, where <i>n</i> is the total number of vertices of all obstacles. Recently, by modifying Hershberger and Suri’s algorithm, Wang (in SODA’21) reduced the space to <i>O(n)</i> while the runtime of the algorithm is still <i>O(n</i> log <i>n</i>). In this article, we present a new algorithm of <i>O(n+h</i> log <i>h</i>) time and <i>O(n)</i> space, provided that a triangulation of the free space is given, where <i>h</i> is the number of obstacles. The algorithm is better than the previous work when <i>h</i> is relatively small. Our algorithm builds a shortest path map for a source point <i>s</i> so that given any query point <i>t</i>, the shortest path length from <i>s</i> to <i>t</i> can be computed in <i>O</i>(log <i>n</i>) time and a shortest <i>s</i>-<i>t</i> path can be produced in additional time linear in the number of edges of the path.</p>","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2023-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138525714","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Bounding the price of anarchy, which quantifies the damage to social welfare due to selfish behavior of the participants, has been an important area of research in algorithmic game theory. Classical work on such bounds in repeated games makes the strong assumption that the subsequent rounds of the repeated games are independent beyond any influence on play from past history. This work studies such bounds in environments that themselves change due to the actions of the agents. Concretely, we consider this problem in discrete-time queuing systems, where competitive queues try to get their packets served. In this model, a queue gets to send a packet at each step to one of the servers, which will attempt to serve the oldest arriving packet, and unprocessed packets are returned to each queue. We model this as a repeated game where queues compete for the capacity of the servers, but where the state of the game evolves as the length of each queue varies. We analyze this queuing system from multiple perspectives. As a baseline measure, we first establish precise conditions on the queuing arrival rates and service capacities that ensure all packets clear efficiently under centralized coordination. We then show that if queues strategically choose servers according to independent and stationary distributions, the system remains stable provided it would be stable under coordination with arrival rates scaled up by a factor of just (frac{e}{e-1}) . Finally, we extend these results to no-regret learning dynamics: if queues use learning algorithms satisfying the no-regret property to choose servers, then the requisite factor increases to 2, and both of these bounds are tight. Both of these results require new probabilistic techniques compared to the classical price of anarchy literature and show that in such settings, no-regret learning can exhibit efficiency loss due to myopia.
{"title":"The Price of Anarchy of Strategic Queuing Systems","authors":"J. Gaitonde, É. Tardos","doi":"10.1145/3587250","DOIUrl":"https://doi.org/10.1145/3587250","url":null,"abstract":"Bounding the price of anarchy, which quantifies the damage to social welfare due to selfish behavior of the participants, has been an important area of research in algorithmic game theory. Classical work on such bounds in repeated games makes the strong assumption that the subsequent rounds of the repeated games are independent beyond any influence on play from past history. This work studies such bounds in environments that themselves change due to the actions of the agents. Concretely, we consider this problem in discrete-time queuing systems, where competitive queues try to get their packets served. In this model, a queue gets to send a packet at each step to one of the servers, which will attempt to serve the oldest arriving packet, and unprocessed packets are returned to each queue. We model this as a repeated game where queues compete for the capacity of the servers, but where the state of the game evolves as the length of each queue varies. We analyze this queuing system from multiple perspectives. As a baseline measure, we first establish precise conditions on the queuing arrival rates and service capacities that ensure all packets clear efficiently under centralized coordination. We then show that if queues strategically choose servers according to independent and stationary distributions, the system remains stable provided it would be stable under coordination with arrival rates scaled up by a factor of just (frac{e}{e-1}) . Finally, we extend these results to no-regret learning dynamics: if queues use learning algorithms satisfying the no-regret property to choose servers, then the requisite factor increases to 2, and both of these bounds are tight. Both of these results require new probabilistic techniques compared to the classical price of anarchy literature and show that in such settings, no-regret learning can exhibit efficiency loss due to myopia.","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2023-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88165109","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Meena Jagadeesan, Alexander Wei, Yixin Wang, Michael I. Jordan, Jacob Steinhardt
Large-scale, two-sided matching platforms must find market outcomes that align with user preferences while simultaneously learning these preferences from data. Classical notions of stability (Gale and Shapley, 1962; Shapley and Shubik, 1971) are, unfortunately, of limited value in the learning setting, given that preferences are inherently uncertain and destabilizing while they are being learned. To bridge this gap, we develop a framework and algorithms for learning stable market outcomes under uncertainty. Our primary setting is matching with transferable utilities, where the platform both matches agents and sets monetary transfers between them. We design an incentive-aware learning objective that captures the distance of a market outcome from equilibrium. Using this objective, we analyze the complexity of learning as a function of preference structure, casting learning as a stochastic multi-armed bandit problem. Algorithmically, we show that “optimism in the face of uncertainty,” the principle underlying many bandit algorithms, applies to a primal-dual formulation of matching with transfers and leads to near-optimal regret bounds. Our work takes a first step toward elucidating when and how stable matchings arise in large, data-driven marketplaces.
大规模的双边匹配平台必须找到符合用户偏好的市场结果,同时从数据中学习这些偏好。稳定性的经典概念(Gale and Shapley, 1962;Shapley和Shubik, 1971)不幸的是,在学习环境中价值有限,因为偏好在学习过程中具有固有的不确定性和不稳定性。为了弥补这一差距,我们开发了一个框架和算法来学习不确定性下的稳定市场结果。我们的主要设置是与可转让的公用事业相匹配,平台既匹配代理,又设置他们之间的货币转移。我们设计了一个激励意识学习目标,捕捉市场结果与均衡的距离。基于这一目标,我们分析了学习的复杂性作为偏好结构的函数,将学习作为一个随机的多臂强盗问题。在算法上,我们展示了“面对不确定性的乐观主义”,这是许多强盗算法的基本原则,适用于与转移匹配的原始对偶公式,并导致接近最优的后悔界限。我们的工作向阐明稳定匹配何时以及如何在大型数据驱动的市场中出现迈出了第一步。
{"title":"Learning Equilibria in Matching Markets with Bandit Feedback","authors":"Meena Jagadeesan, Alexander Wei, Yixin Wang, Michael I. Jordan, Jacob Steinhardt","doi":"10.1145/3583681","DOIUrl":"https://doi.org/10.1145/3583681","url":null,"abstract":"Large-scale, two-sided matching platforms must find market outcomes that align with user preferences while simultaneously learning these preferences from data. Classical notions of stability (Gale and Shapley, 1962; Shapley and Shubik, 1971) are, unfortunately, of limited value in the learning setting, given that preferences are inherently uncertain and destabilizing while they are being learned. To bridge this gap, we develop a framework and algorithms for learning stable market outcomes under uncertainty. Our primary setting is matching with transferable utilities, where the platform both matches agents and sets monetary transfers between them. We design an incentive-aware learning objective that captures the distance of a market outcome from equilibrium. Using this objective, we analyze the complexity of learning as a function of preference structure, casting learning as a stochastic multi-armed bandit problem. Algorithmically, we show that “optimism in the face of uncertainty,” the principle underlying many bandit algorithms, applies to a primal-dual formulation of matching with transfers and leads to near-optimal regret bounds. Our work takes a first step toward elucidating when and how stable matchings arise in large, data-driven marketplaces.","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2023-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83992234","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work represents a natural coalescence of two important lines of work — learning mixtures of Gaussians and algorithmic robust statistics. In particular, we give the first provably robust algorithm for learning mixtures of any constant number of Gaussians. We require only mild assumptions on the mixing weights and that the total variation distance between components is bounded away from zero. At the heart of our algorithm is a new method for proving a type of dimension-independent polynomial identifiability — which we call robust identifiability — through applying a carefully chosen sequence of differential operations to certain generating functions that not only encode the parameters we would like to learn but also the system of polynomial equations we would like to solve. We show how the symbolic identities we derive can be directly used to analyze a natural sum-of-squares relaxation.
{"title":"Robustly Learning General Mixtures of Gaussians","authors":"Allen Liu, Ankur Moitra","doi":"10.1145/3583680","DOIUrl":"https://doi.org/10.1145/3583680","url":null,"abstract":"This work represents a natural coalescence of two important lines of work — learning mixtures of Gaussians and algorithmic robust statistics. In particular, we give the first provably robust algorithm for learning mixtures of any constant number of Gaussians. We require only mild assumptions on the mixing weights and that the total variation distance between components is bounded away from zero. At the heart of our algorithm is a new method for proving a type of dimension-independent polynomial identifiability — which we call robust identifiability — through applying a carefully chosen sequence of differential operations to certain generating functions that not only encode the parameters we would like to learn but also the system of polynomial equations we would like to solve. We show how the symbolic identities we derive can be directly used to analyze a natural sum-of-squares relaxation.","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2023-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79878742","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
R. Bruni, R. Giacobazzi, R. Gori, Francesco Ranzato
Abstract interpretation is a well-known and extensively used method to extract over-approximate program invariants by a sound program analysis algorithm. Soundness means that no program errors are lost and it is, in principle, guaranteed by construction. Completeness means that the abstract interpreter reports no false alarms for all possible inputs, but this is extremely rare because it needs a very precise analysis. We introduce a weaker notion of completeness, called local completeness, which requires that no false alarms are produced only relatively to some fixed program inputs. Based on this idea, we introduce a program logic, called Local Completeness Logic for an abstract domain A, for proving both the correctness and incorrectness of program specifications. Our proof system, which is parameterized by an abstract domain A, combines over- and under-approximating reasoning. In a provable triple ⊦A [p] 𝖼 [q], 𝖼 is a program, q is an under-approximation of the strongest post-condition of 𝖼 on input p such that their abstractions in A coincide. This means that q is never too coarse, namely, under some mild assumptions, the abstract interpretation of 𝖼 does not yield false alarms for the input p iff q has no alarm. Therefore, proving ⊦A [p] 𝖼 [q] not only ensures that all the alarms raised in q are true ones, but also that if q does not raise alarms, then 𝖼 is correct. We also prove that if A is the straightforward abstraction making all program properties equivalent, then our program logic coincides with O’Hearn’s incorrectness logic, while for any other abstraction, contrary to the case of incorrectness logic, our logic can also establish program correctness.
{"title":"A Correctness and Incorrectness Program Logic","authors":"R. Bruni, R. Giacobazzi, R. Gori, Francesco Ranzato","doi":"10.1145/3582267","DOIUrl":"https://doi.org/10.1145/3582267","url":null,"abstract":"Abstract interpretation is a well-known and extensively used method to extract over-approximate program invariants by a sound program analysis algorithm. Soundness means that no program errors are lost and it is, in principle, guaranteed by construction. Completeness means that the abstract interpreter reports no false alarms for all possible inputs, but this is extremely rare because it needs a very precise analysis. We introduce a weaker notion of completeness, called local completeness, which requires that no false alarms are produced only relatively to some fixed program inputs. Based on this idea, we introduce a program logic, called Local Completeness Logic for an abstract domain A, for proving both the correctness and incorrectness of program specifications. Our proof system, which is parameterized by an abstract domain A, combines over- and under-approximating reasoning. In a provable triple ⊦A [p] 𝖼 [q], 𝖼 is a program, q is an under-approximation of the strongest post-condition of 𝖼 on input p such that their abstractions in A coincide. This means that q is never too coarse, namely, under some mild assumptions, the abstract interpretation of 𝖼 does not yield false alarms for the input p iff q has no alarm. Therefore, proving ⊦A [p] 𝖼 [q] not only ensures that all the alarms raised in q are true ones, but also that if q does not raise alarms, then 𝖼 is correct. We also prove that if A is the straightforward abstraction making all program properties equivalent, then our program logic coincides with O’Hearn’s incorrectness logic, while for any other abstraction, contrary to the case of incorrectness logic, our logic can also establish program correctness.","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2023-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73583749","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-16DOI: https://dl.acm.org/doi/10.1145/3570637
Nai-Hui Chia, Kai-Min Chung, Ching-Yi Lai
Near-term quantum computers are likely to have small depths due to short coherence time and noisy gates. A natural approach to leverage these quantum computers is interleaving them with classical computers. Understanding the capabilities and limits of this hybrid approach is an essential topic in quantum computation. Most notably, the quantum Fourier transform can be implemented by a hybrid of logarithmic-depth quantum circuits and a classical polynomial-time algorithm. Therefore, it seems possible that quantum polylogarithmic depth is as powerful as quantum polynomial depth in the presence of classical computation. Indeed, Jozsa conjectured that “Any quantum polynomial-time algorithm can be implemented with only O(log n) quantum depth interspersed with polynomial-time classical computations.” This can be formalized as asserting the equivalence of BQP and “BQNCBPP.” However, Aaronson conjectured that “there exists an oracle separation between BQP and BPPBQNC.” BQNCBPP and BPPBQNC are two natural and seemingly incomparable ways of hybrid classical-quantum computation.
In this work, we manage to prove Aaronson’s conjecture and in the meantime prove that Jozsa’s conjecture, relative to an oracle, is false. In fact, we prove a stronger statement that for any depth parameter d, there exists an oracle that separates quantum depth d and 2d+1 in the presence of classical computation. Thus, our results show that relative to oracles, doubling the quantum circuit depth does make the hybrid model more powerful, and this cannot be traded by classical computation.
{"title":"On the Need for Large Quantum Depth","authors":"Nai-Hui Chia, Kai-Min Chung, Ching-Yi Lai","doi":"https://dl.acm.org/doi/10.1145/3570637","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3570637","url":null,"abstract":"<p>Near-term quantum computers are likely to have small depths due to short coherence time and noisy gates. A natural approach to leverage these quantum computers is interleaving them with classical computers. Understanding the capabilities and limits of this hybrid approach is an essential topic in quantum computation. Most notably, the quantum Fourier transform can be implemented by a hybrid of logarithmic-depth quantum circuits and a classical polynomial-time algorithm. Therefore, it seems possible that quantum polylogarithmic depth is as powerful as quantum polynomial depth in the presence of classical computation. Indeed, Jozsa conjectured that “<i>Any quantum polynomial-time algorithm can be implemented with only <i>O</i>(log <i>n</i>) quantum depth interspersed with polynomial-time classical computations.</i>” This can be formalized as asserting the equivalence of <monospace>BQP</monospace> and “<monospace>BQNC<sup>BPP</sup></monospace>.” However, Aaronson conjectured that “<i>there exists an oracle separation between <monospace>BQP</monospace> and <monospace>BPP<sup>BQNC</sup></monospace>.</i>” <monospace>BQNC<sup>BPP</sup></monospace> and <monospace>BPP<sup>BQNC</sup></monospace> are two natural and seemingly incomparable ways of hybrid classical-quantum computation.</p><p>In this work, we manage to prove Aaronson’s conjecture and in the meantime prove that Jozsa’s conjecture, relative to an oracle, is false. In fact, we prove a stronger statement that for any depth parameter <i>d</i>, there exists an oracle that separates quantum depth <i>d</i> and 2<i>d</i>+1 in the presence of classical computation. Thus, our results show that relative to oracles, doubling the quantum circuit depth does make the hybrid model more powerful, and this cannot be traded by classical computation.</p>","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2023-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138525672","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-16DOI: https://dl.acm.org/doi/10.1145/3570636
Xavier Goaoc, Emo Welzl
We establish the following two main results on order types of points in general position in the plane (realizable simple planar order types, realizable uniform acyclic oriented matroids of rank 3):
(a)
The number of extreme points in an n-point order type, chosen uniformly at random from all such order types, is on average 4+o(1). For labeled order types, this number has average (4- mbox{$frac{8}{n^2 - n +2}$}) and variance at most 3.
(b)
The (labeled) order types read off a set of n points sampled independently from the uniform measure on a convex planar domain, smooth or polygonal, or from a Gaussian distribution are concentrated, i.e., such sampling typically encounters only a vanishingly small fraction of all order types of the given size.
Result (a) generalizes to arbitrary dimension d for labeled order types with the average number of extreme points 2d+o (1) and constant variance. We also discuss to what extent our methods generalize to the abstract setting of uniform acyclic oriented matroids. Moreover, our methods show the following relative of the Erdős-Szekeres theorem: for any fixed k, as n → ∞, a proportion 1 - O(1/n) of the n-point simple order types contain a triangle enclosing a convex k-chain over an edge.
For the unlabeled case in (a), we prove that for any antipodal, finite subset of the two-dimensional sphere, the group of orientation preserving bijections is cyclic, dihedral, or one of A4, S4, or A5 (and each case is possible). These are the finite subgroups of SO(3) and our proof follows the lines of their characterization by Felix Klein.
我们建立了平面上一般位置点的阶型(可实现的简单平面阶型,可实现的秩为3的一致无环定向阵)的两个主要结果:(a)从所有这些阶型中均匀随机选择的n点阶型的极值点的数目平均为4+o(1)。对于标记的阶型,这个数字的平均值为(4- mbox{$frac{8}{n^2 - n +2}$}),方差最多为3。(b)从凸平面域、光滑或多边形或高斯分布的均匀测量中独立采样的n个点的集合中读取(标记的)阶型是集中的,即,这种采样通常只遇到给定大小的所有阶型的一小部分。结果(a)推广到任意维度d,对于有标记的阶型,极值点的平均个数为2d+o(1),方差为常数。我们还讨论了我们的方法在多大程度上推广到一致无环定向拟阵的抽象集合。此外,我们的方法还证明了Erdős-Szekeres定理的如下关系:对于任意固定k,当n→∞时,n点简单阶型的比例1 - O(1/n)包含一个在边上包有凸k链的三角形。对于(a)中的未标记情况,我们证明了对于二维球面的任何对映有限子集,保向双射群是循环的、二面体的或A4、S4、A5中的一个(每种情况都是可能的)。这些是SO(3)的有限子群,我们的证明遵循了Felix Klein对它们的描述。
{"title":"Convex Hulls of Random Order Types","authors":"Xavier Goaoc, Emo Welzl","doi":"https://dl.acm.org/doi/10.1145/3570636","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3570636","url":null,"abstract":"<p>We establish the following two main results on order types of points in general position in the plane (realizable simple planar order types, realizable uniform acyclic oriented matroids of rank 3):\u0000<p><table border=\"0\" list-type=\"ordered\" width=\"95%\"><tr><td valign=\"top\"><p>(a)</p></td><td colspan=\"5\" valign=\"top\"><p>The number of extreme points in an <i>n</i>-point order type, chosen uniformly at random from all such order types, is on average 4+<i>o</i>(1). For labeled order types, this number has average (4- mbox{$frac{8}{n^2 - n +2}$}) and variance at most 3.</p></td></tr><tr><td valign=\"top\"><p>(b)</p></td><td colspan=\"5\" valign=\"top\"><p>The (labeled) order types read off a set of <i>n</i> points sampled independently from the uniform measure on a convex planar domain, smooth or polygonal, or from a Gaussian distribution are concentrated, i.e., such sampling typically encounters only a vanishingly small fraction of all order types of the given size.</p></td></tr></table></p> Result (a) generalizes to arbitrary dimension <i>d</i> for labeled order types with the average number of extreme points 2<i>d</i>+<i>o</i> (1) and constant variance. We also discuss to what extent our methods generalize to the abstract setting of uniform acyclic oriented matroids. Moreover, our methods show the following relative of the Erdős-Szekeres theorem: for any fixed <i>k</i>, as <i>n</i> → ∞, a proportion 1 - <i>O</i>(1/<i>n</i>) of the <i>n</i>-point simple order types contain a triangle enclosing a convex <i>k</i>-chain over an edge.</p><p>For the unlabeled case in (a), we prove that for any antipodal, finite subset of the two-dimensional sphere, the group of orientation preserving bijections is cyclic, dihedral, or one of <i>A</i><sub>4</sub>, <i>S</i><sub>4</sub>, or <i>A</i><sub>5</sub> (and each case is possible). These are the finite subgroups of <i>SO</i>(3) and our proof follows the lines of their characterization by Felix Klein.</p>","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2023-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138525688","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abraham, Dolev, Geffner, and Halpern [1] proved that, in asynchronous systems, a (k, t)-robust equilibrium for n players and a trusted mediator can be implemented without the mediator as long as n > 4(k+t), where an equilibrium is (k, t)-robust if, roughly speaking, no coalition of t players can decrease the payoff of any of the other players, and no coalition of k players can increase their payoff by deviating. We prove that this bound is tight, in the sense that if n ≤ 4(k+t) there exist (k, t)-robust equilibria with a mediator that cannot be implemented by the players alone. Even though implementing (k, t)-robust mediators seems closely related to implementing asynchronous multiparty (k+t)-secure computation [6], to the best of our knowledge there is no known straightforward reduction from one problem to another. Nevertheless, we show that there is a non-trivial reduction from a slightly weaker notion of (k+t)-secure computation, which we call (k+t)-strict secure computation, to implementing (k, t)-robust mediators. We prove the desired lower bound by showing that there are functions on n variables that cannot be (k+t)-strictly securely computed if n ≤ 4(k+t). This also provides a simple alternative proof for the well-known lower bound of 4t+1 on asynchronous secure computation in the presence of up to t malicious agents [4, 8, 10].
{"title":"Lower Bounds on Implementing Mediators in Asynchronous Systems with Rational and Malicious Agents","authors":"I. Geffner, Joseph Y. Halpern","doi":"10.1145/3578579","DOIUrl":"https://doi.org/10.1145/3578579","url":null,"abstract":"Abraham, Dolev, Geffner, and Halpern [1] proved that, in asynchronous systems, a (k, t)-robust equilibrium for n players and a trusted mediator can be implemented without the mediator as long as n > 4(k+t), where an equilibrium is (k, t)-robust if, roughly speaking, no coalition of t players can decrease the payoff of any of the other players, and no coalition of k players can increase their payoff by deviating. We prove that this bound is tight, in the sense that if n ≤ 4(k+t) there exist (k, t)-robust equilibria with a mediator that cannot be implemented by the players alone. Even though implementing (k, t)-robust mediators seems closely related to implementing asynchronous multiparty (k+t)-secure computation [6], to the best of our knowledge there is no known straightforward reduction from one problem to another. Nevertheless, we show that there is a non-trivial reduction from a slightly weaker notion of (k+t)-secure computation, which we call (k+t)-strict secure computation, to implementing (k, t)-robust mediators. We prove the desired lower bound by showing that there are functions on n variables that cannot be (k+t)-strictly securely computed if n ≤ 4(k+t). This also provides a simple alternative proof for the well-known lower bound of 4t+1 on asynchronous secure computation in the presence of up to t malicious agents [4, 8, 10].","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2023-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80073995","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-19DOI: https://dl.acm.org/doi/10.1145/3568163
John Fearnley, Paul Goldberg, Alexandros Hollender, Rahul Savani
We study search problems that can be solved by performing Gradient Descent on a bounded convex polytopal domain and show that this class is equal to the intersection of two well-known classes: PPAD and PLS. As our main underlying technical contribution, we show that computing a Karush-Kuhn-Tucker (KKT) point of a continuously differentiable function over the domain [0,1]2 is PPAD ∩ PLS-complete. This is the first non-artificial problem to be shown complete for this class. Our results also imply that the class CLS (Continuous Local Search) – which was defined by Daskalakis and Papadimitriou as a more “natural” counterpart to PPAD ∩ PLS and contains many interesting problems – is itself equal to PPAD ∩ PLS.
{"title":"The Complexity of Gradient Descent: CLS = PPAD ∩ PLS","authors":"John Fearnley, Paul Goldberg, Alexandros Hollender, Rahul Savani","doi":"https://dl.acm.org/doi/10.1145/3568163","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3568163","url":null,"abstract":"<p>We study search problems that can be solved by performing Gradient Descent on a bounded convex polytopal domain and show that this class is equal to the intersection of two well-known classes: PPAD and PLS. As our main underlying technical contribution, we show that computing a Karush-Kuhn-Tucker (KKT) point of a continuously differentiable function over the domain [0,1]<sup>2</sup> is PPAD ∩ PLS-complete. This is the first non-artificial problem to be shown complete for this class. Our results also imply that the class CLS (Continuous Local Search) – which was defined by Daskalakis and Papadimitriou as a more “natural” counterpart to PPAD ∩ PLS and contains many interesting problems – is itself equal to PPAD ∩ PLS.</p>","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2022-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138525665","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-19DOI: https://dl.acm.org/doi/10.1145/3566051
Manuel Bodirsky, Jakub Rydval
Finite-domain constraint satisfaction problems are either solvable by Datalog or not even expressible in fixed-point logic with counting. The border between the two regimes can be described by a universal-algebraic minor condition. For infinite-domain constraint satisfaction problems (CSPs), the situation is more complicated even if the template structure of the CSP is model-theoretically tame. We prove that there is no Maltsev condition that characterizes Datalog already for the CSPs of first-order reducts of (ℚ;<); such CSPs are called temporal CSPs and are of fundamental importance in infinite-domain constraint satisfaction. Our main result is a complete classification of temporal CSPs that can be expressed in one of the following logical formalisms: Datalog, fixed-point logic (with or without counting), or fixed-point logic with the mod-2 rank operator. The classification shows that many of the equivalent conditions in the finite fail to capture expressibility in Datalog or fixed-point logic already for temporal CSPs.
{"title":"On the Descriptive Complexity of Temporal Constraint Satisfaction Problems","authors":"Manuel Bodirsky, Jakub Rydval","doi":"https://dl.acm.org/doi/10.1145/3566051","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3566051","url":null,"abstract":"<p>Finite-domain constraint satisfaction problems are either solvable by Datalog or not even expressible in fixed-point logic with counting. The border between the two regimes can be described by a universal-algebraic minor condition. For infinite-domain constraint satisfaction problems (CSPs), the situation is more complicated even if the template structure of the CSP is model-theoretically tame. We prove that there is no Maltsev condition that characterizes Datalog already for the CSPs of first-order reducts of (ℚ;<); such CSPs are called <i>temporal CSPs</i> and are of fundamental importance in infinite-domain constraint satisfaction. Our main result is a complete classification of temporal CSPs that can be expressed in one of the following logical formalisms: Datalog, fixed-point logic (with or without counting), or fixed-point logic with the mod-2 rank operator. The classification shows that many of the equivalent conditions in the finite fail to capture expressibility in Datalog or fixed-point logic already for temporal CSPs.</p>","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2022-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138525667","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}