Pub Date : 2022-07-18DOI: 10.1137/1.9781611977554.ch46
A. Cornelissen, Yassine Hamoudi
We present a novel quantum algorithm for estimating Gibbs partition functions in sublinear time with respect to the logarithm of the size of the state space. This is the first speed-up of this type to be obtained over the seminal nearly-linear time algorithm of v{S}tefankoviv{c}, Vempala and Vigoda [JACM, 2009]. Our result also preserves the quadratic speed-up in precision and spectral gap achieved in previous work by exploiting the properties of quantum Markov chains. As an application, we obtain new polynomial improvements over the best-known algorithms for computing the partition function of the Ising model, counting the number of $k$-colorings, matchings or independent sets of a graph, and estimating the volume of a convex body. Our approach relies on developing new variants of the quantum phase and amplitude estimation algorithms that return nearly unbiased estimates with low variance and without destroying their initial quantum state. We extend these subroutines into a nearly unbiased quantum mean estimator that reduces the variance quadratically faster than the classical empirical mean. No such estimator was known to exist prior to our work. These properties, which are of general interest, lead to better convergence guarantees within the paradigm of simulated annealing for computing partition functions.
{"title":"A Sublinear-Time Quantum Algorithm for Approximating Partition Functions","authors":"A. Cornelissen, Yassine Hamoudi","doi":"10.1137/1.9781611977554.ch46","DOIUrl":"https://doi.org/10.1137/1.9781611977554.ch46","url":null,"abstract":"We present a novel quantum algorithm for estimating Gibbs partition functions in sublinear time with respect to the logarithm of the size of the state space. This is the first speed-up of this type to be obtained over the seminal nearly-linear time algorithm of v{S}tefankoviv{c}, Vempala and Vigoda [JACM, 2009]. Our result also preserves the quadratic speed-up in precision and spectral gap achieved in previous work by exploiting the properties of quantum Markov chains. As an application, we obtain new polynomial improvements over the best-known algorithms for computing the partition function of the Ising model, counting the number of $k$-colorings, matchings or independent sets of a graph, and estimating the volume of a convex body. Our approach relies on developing new variants of the quantum phase and amplitude estimation algorithms that return nearly unbiased estimates with low variance and without destroying their initial quantum state. We extend these subroutines into a nearly unbiased quantum mean estimator that reduces the variance quadratically faster than the classical empirical mean. No such estimator was known to exist prior to our work. These properties, which are of general interest, lead to better convergence guarantees within the paradigm of simulated annealing for computing partition functions.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"1 1","pages":"1245-1264"},"PeriodicalIF":0.0,"publicationDate":"2022-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90323000","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-18DOI: 10.48550/arXiv.2207.08800
Joran van Apeldoorn, A. Cornelissen, Andr'as Gily'en, G. Nannicini
We describe algorithms to obtain an approximate classical description of a $d$-dimensional quantum state when given access to a unitary (and its inverse) that prepares it. For pure states we characterize the query complexity for $ell_q$-norm error up to logarithmic factors. As a special case, we show that it takes $widetilde{Theta}(d/varepsilon)$ applications of the unitaries to obtain an $varepsilon$-$ell_2$-approximation of the state. For mixed states we consider a similar model, where the unitary prepares a purification of the state. In this model we give an efficient algorithm for obtaining Schatten $q$-norm estimates of a rank-$r$ mixed state, giving query upper bounds that are close to optimal. In particular, we show that a trace-norm ($q=1$) estimate can be obtained with $widetilde{mathcal{O}}(dr/varepsilon)$ queries. This improves (assuming our stronger input model) the $varepsilon$-dependence over the algorithm of Haah et al. (2017) that uses a joint measurement on $widetilde{mathcal{O}}(dr/varepsilon^2)$ copies of the state. To our knowledge, the most sample-efficient results for pure-state tomography come from setting the rank to $1$ in generic mixed-state tomography algorithms, which can be computationally demanding. We describe sample-optimal algorithms for pure states that are easy and fast to implement. Along the way we show that an $ell_infty$-norm estimate of a normalized vector induces a (slightly worse) $ell_q$-norm estimate for that vector, without losing a dimension-dependent factor in the precision. We also develop an unbiased and symmetric version of phase estimation, where the probability distribution of the estimate is centered around the true value. Finally, we give an efficient method for estimating multiple expectation values, improving over the recent result by Huggins et al. (2021) when the measurement operators do not fully overlap.
{"title":"Quantum tomography using state-preparation unitaries","authors":"Joran van Apeldoorn, A. Cornelissen, Andr'as Gily'en, G. Nannicini","doi":"10.48550/arXiv.2207.08800","DOIUrl":"https://doi.org/10.48550/arXiv.2207.08800","url":null,"abstract":"We describe algorithms to obtain an approximate classical description of a $d$-dimensional quantum state when given access to a unitary (and its inverse) that prepares it. For pure states we characterize the query complexity for $ell_q$-norm error up to logarithmic factors. As a special case, we show that it takes $widetilde{Theta}(d/varepsilon)$ applications of the unitaries to obtain an $varepsilon$-$ell_2$-approximation of the state. For mixed states we consider a similar model, where the unitary prepares a purification of the state. In this model we give an efficient algorithm for obtaining Schatten $q$-norm estimates of a rank-$r$ mixed state, giving query upper bounds that are close to optimal. In particular, we show that a trace-norm ($q=1$) estimate can be obtained with $widetilde{mathcal{O}}(dr/varepsilon)$ queries. This improves (assuming our stronger input model) the $varepsilon$-dependence over the algorithm of Haah et al. (2017) that uses a joint measurement on $widetilde{mathcal{O}}(dr/varepsilon^2)$ copies of the state. To our knowledge, the most sample-efficient results for pure-state tomography come from setting the rank to $1$ in generic mixed-state tomography algorithms, which can be computationally demanding. We describe sample-optimal algorithms for pure states that are easy and fast to implement. Along the way we show that an $ell_infty$-norm estimate of a normalized vector induces a (slightly worse) $ell_q$-norm estimate for that vector, without losing a dimension-dependent factor in the precision. We also develop an unbiased and symmetric version of phase estimation, where the probability distribution of the estimate is centered around the true value. Finally, we give an efficient method for estimating multiple expectation values, improving over the recent result by Huggins et al. (2021) when the measurement operators do not fully overlap.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"41 1","pages":"1265-1318"},"PeriodicalIF":0.0,"publicationDate":"2022-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82361014","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-18DOI: 10.48550/arXiv.2207.08347
Sivakanth Gopi, Y. Lee, Daogao Liu, Ruoqi Shen, Kevin Tian
We propose a new framework for differentially private optimization of convex functions which are Lipschitz in an arbitrary norm $|cdot|$. Our algorithms are based on a regularized exponential mechanism which samples from the density $propto exp(-k(F+mu r))$ where $F$ is the empirical loss and $r$ is a regularizer which is strongly convex with respect to $|cdot|$, generalizing a recent work of [Gopi, Lee, Liu '22] to non-Euclidean settings. We show that this mechanism satisfies Gaussian differential privacy and solves both DP-ERM (empirical risk minimization) and DP-SCO (stochastic convex optimization) by using localization tools from convex geometry. Our framework is the first to apply to private convex optimization in general normed spaces and directly recovers non-private SCO rates achieved by mirror descent as the privacy parameter $epsilon to infty$. As applications, for Lipschitz optimization in $ell_p$ norms for all $p in (1, 2)$, we obtain the first optimal privacy-utility tradeoffs; for $p = 1$, we improve tradeoffs obtained by the recent works [Asi, Feldman, Koren, Talwar '21, Bassily, Guzman, Nandi '21] by at least a logarithmic factor. Our $ell_p$ norm and Schatten-$p$ norm optimization frameworks are complemented with polynomial-time samplers whose query complexity we explicitly bound.
我们提出了一个新的框架来求解任意范数Lipschitz凸函数的微分私有优化$|cdot|$。我们的算法基于正则化指数机制，该机制从密度$propto exp(-k(F+mu r))$中采样，其中$F$是经验损失，$r$是正则化器，相对于$|cdot|$是强凸的，将[Gopi, Lee, Liu '22]的最新工作推广到非欧几里得设置。我们证明了这种机制满足高斯微分隐私，并通过使用凸几何的定位工具解决了DP-ERM(经验风险最小化)和DP-SCO(随机凸优化)。我们的框架是第一个应用于一般赋范空间中的私有凸优化，并直接恢复通过镜像下降作为隐私参数$epsilon to infty$实现的非私有SCO率。作为应用，对于Lipschitz优化在$ell_p$规范对所有$p in (1, 2)$，我们得到了第一个最优的隐私效用权衡;对于$p = 1$，我们至少通过对数因子改进了最近的作品[Asi, Feldman, Koren, Talwar '21, Bassily, Guzman, Nandi '21]所获得的权衡。我们的$ell_p$范数和Schatten- $p$范数优化框架补充了多项式时间采样器，其查询复杂度我们明确地限定了。
{"title":"Private Convex Optimization in General Norms","authors":"Sivakanth Gopi, Y. Lee, Daogao Liu, Ruoqi Shen, Kevin Tian","doi":"10.48550/arXiv.2207.08347","DOIUrl":"https://doi.org/10.48550/arXiv.2207.08347","url":null,"abstract":"We propose a new framework for differentially private optimization of convex functions which are Lipschitz in an arbitrary norm $|cdot|$. Our algorithms are based on a regularized exponential mechanism which samples from the density $propto exp(-k(F+mu r))$ where $F$ is the empirical loss and $r$ is a regularizer which is strongly convex with respect to $|cdot|$, generalizing a recent work of [Gopi, Lee, Liu '22] to non-Euclidean settings. We show that this mechanism satisfies Gaussian differential privacy and solves both DP-ERM (empirical risk minimization) and DP-SCO (stochastic convex optimization) by using localization tools from convex geometry. Our framework is the first to apply to private convex optimization in general normed spaces and directly recovers non-private SCO rates achieved by mirror descent as the privacy parameter $epsilon to infty$. As applications, for Lipschitz optimization in $ell_p$ norms for all $p in (1, 2)$, we obtain the first optimal privacy-utility tradeoffs; for $p = 1$, we improve tradeoffs obtained by the recent works [Asi, Feldman, Koren, Talwar '21, Bassily, Guzman, Nandi '21] by at least a logarithmic factor. Our $ell_p$ norm and Schatten-$p$ norm optimization frameworks are complemented with polynomial-time samplers whose query complexity we explicitly bound.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"39 1","pages":"5068-5089"},"PeriodicalIF":0.0,"publicationDate":"2022-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75885533","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-18DOI: 10.48550/arXiv.2207.08783
Haim Kaplan, David Naori, D. Raz
We study the online facility location problem with uniform facility costs in the random-order model. Meyerson's algorithm [FOCS'01] is arguably the most natural and simple online algorithm for the problem with several advantages and appealing properties. Its analysis in the random-order model is one of the cornerstones of random-order analysis beyond the secretary problem. Meyerson's algorithm was shown to be (asymptotically) optimal in the standard worst-case adversarial-order model and $8$-competitive in the random order model. While this bound in the random-order model is the long-standing state-of-the-art, it is not known to be tight, and the true competitive-ratio of Meyerson's algorithm remained an open question for more than two decades. We resolve this question and prove tight bounds on the competitive-ratio of Meyerson's algorithm in the random-order model, showing that it is exactly $4$-competitive. Following our tight analysis, we introduce a generic parameterized version of Meyerson's algorithm that retains all the advantages of the original version. We show that the best algorithm in this family is exactly $3$-competitive. On the other hand, we show that no online algorithm for this problem can achieve a competitive-ratio better than $2$. Finally, we prove that the algorithms in this family are robust to partial adversarial arrival orders.
{"title":"Almost Tight Bounds for Online Facility Location in the Random-Order Model","authors":"Haim Kaplan, David Naori, D. Raz","doi":"10.48550/arXiv.2207.08783","DOIUrl":"https://doi.org/10.48550/arXiv.2207.08783","url":null,"abstract":"We study the online facility location problem with uniform facility costs in the random-order model. Meyerson's algorithm [FOCS'01] is arguably the most natural and simple online algorithm for the problem with several advantages and appealing properties. Its analysis in the random-order model is one of the cornerstones of random-order analysis beyond the secretary problem. Meyerson's algorithm was shown to be (asymptotically) optimal in the standard worst-case adversarial-order model and $8$-competitive in the random order model. While this bound in the random-order model is the long-standing state-of-the-art, it is not known to be tight, and the true competitive-ratio of Meyerson's algorithm remained an open question for more than two decades. We resolve this question and prove tight bounds on the competitive-ratio of Meyerson's algorithm in the random-order model, showing that it is exactly $4$-competitive. Following our tight analysis, we introduce a generic parameterized version of Meyerson's algorithm that retains all the advantages of the original version. We show that the best algorithm in this family is exactly $3$-competitive. On the other hand, we show that no online algorithm for this problem can achieve a competitive-ratio better than $2$. Finally, we prove that the algorithms in this family are robust to partial adversarial arrival orders.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"10 1","pages":"1523-1544"},"PeriodicalIF":0.0,"publicationDate":"2022-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88280707","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-17DOI: 10.48550/arXiv.2207.08268
David P. Woodruff, T. Yasuda
The seminal work of Cohen and Peng introduced Lewis weight sampling to the theoretical computer science community, yielding fast row sampling algorithms for approximating $d$-dimensional subspaces of $ell_p$ up to $(1+epsilon)$ error. Several works have extended this important primitive to other settings, including the online coreset and sliding window models. However, these results are only for $pin{1,2}$, and results for $p=1$ require a suboptimal $tilde O(d^2/epsilon^2)$ samples. In this work, we design the first nearly optimal $ell_p$ subspace embeddings for all $pin(0,infty)$ in the online coreset and sliding window models. In both models, our algorithms store $tilde O(d^{1lor(p/2)}/epsilon^2)$ rows. This answers a substantial generalization of the main open question of [BDMMUWZ2020], and gives the first results for all $pnotin{1,2}$. Towards our result, we give the first analysis of"one-shot'' Lewis weight sampling of sampling rows proportionally to their Lewis weights, with sample complexity $tilde O(d^{p/2}/epsilon^2)$ for $p>2$. Previously, this scheme was only known to have sample complexity $tilde O(d^{p/2}/epsilon^5)$, whereas $tilde O(d^{p/2}/epsilon^2)$ is known if a more sophisticated recursive sampling is used. The recursive sampling cannot be implemented online, thus necessitating an analysis of one-shot Lewis weight sampling. Our analysis uses a novel connection to online numerical linear algebra. As an application, we obtain the first one-pass streaming coreset algorithms for $(1+epsilon)$ approximation of important generalized linear models, such as logistic regression and $p$-probit regression. Our upper bounds are parameterized by a complexity parameter $mu$ introduced by [MSSW2018], and we show the first lower bounds showing that a linear dependence on $mu$ is necessary.
{"title":"Online Lewis Weight Sampling","authors":"David P. Woodruff, T. Yasuda","doi":"10.48550/arXiv.2207.08268","DOIUrl":"https://doi.org/10.48550/arXiv.2207.08268","url":null,"abstract":"The seminal work of Cohen and Peng introduced Lewis weight sampling to the theoretical computer science community, yielding fast row sampling algorithms for approximating $d$-dimensional subspaces of $ell_p$ up to $(1+epsilon)$ error. Several works have extended this important primitive to other settings, including the online coreset and sliding window models. However, these results are only for $pin{1,2}$, and results for $p=1$ require a suboptimal $tilde O(d^2/epsilon^2)$ samples. In this work, we design the first nearly optimal $ell_p$ subspace embeddings for all $pin(0,infty)$ in the online coreset and sliding window models. In both models, our algorithms store $tilde O(d^{1lor(p/2)}/epsilon^2)$ rows. This answers a substantial generalization of the main open question of [BDMMUWZ2020], and gives the first results for all $pnotin{1,2}$. Towards our result, we give the first analysis of\"one-shot'' Lewis weight sampling of sampling rows proportionally to their Lewis weights, with sample complexity $tilde O(d^{p/2}/epsilon^2)$ for $p>2$. Previously, this scheme was only known to have sample complexity $tilde O(d^{p/2}/epsilon^5)$, whereas $tilde O(d^{p/2}/epsilon^2)$ is known if a more sophisticated recursive sampling is used. The recursive sampling cannot be implemented online, thus necessitating an analysis of one-shot Lewis weight sampling. Our analysis uses a novel connection to online numerical linear algebra. As an application, we obtain the first one-pass streaming coreset algorithms for $(1+epsilon)$ approximation of important generalized linear models, such as logistic regression and $p$-probit regression. Our upper bounds are parameterized by a complexity parameter $mu$ introduced by [MSSW2018], and we show the first lower bounds showing that a linear dependence on $mu$ is necessary.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"30 1","pages":"4622-4666"},"PeriodicalIF":0.0,"publicationDate":"2022-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82668774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-17DOI: 10.48550/arXiv.2207.08311
Cole Franks, Tasuku Soma, M. Goemans
A recent breakthrough in Edmonds' problem showed that the noncommutative rank can be computed in deterministic polynomial time, and various algorithms for it were devised. However, only quite complicated algorithms are known for finding a so-called shrunk subspace, which acts as a dual certificate for the value of the noncommutative rank. In particular, the operator Sinkhorn algorithm, perhaps the simplest algorithm to compute the noncommutative rank with operator scaling, does not find a shrunk subspace. Finding a shrunk subspace plays a key role in applications, such as separation in the Brascamp-Lieb polytope, one-parameter subgroups in the null-cone membership problem, and primal-dual algorithms for matroid intersection and fractional matroid matching. In this paper, we provide a simple Sinkhorn-style algorithm to find the smallest shrunk subspace over the complex field in deterministic polynomial time. To this end, we introduce a generalization of the operator scaling problem, where the spectra of the marginals must be majorized by specified vectors. Then we design an efficient Sinkhorn-style algorithm for the generalized operator scaling problem. Applying this to the shrunk subspace problem, we show that a sufficiently long run of the algorithm also finds an approximate shrunk subspace close to the minimum exact shrunk subspace. Finally, we show that the approximate shrunk subspace can be rounded if it is sufficiently close. Along the way, we also provide a simple randomized algorithm to find the smallest shrunk subspace. As applications, we design a faster algorithm for fractional linear matroid matching and efficient weak membership and optimization algorithms for the rank-2 Brascamp-Lieb polytope.
{"title":"Shrunk subspaces via operator Sinkhorn iteration","authors":"Cole Franks, Tasuku Soma, M. Goemans","doi":"10.48550/arXiv.2207.08311","DOIUrl":"https://doi.org/10.48550/arXiv.2207.08311","url":null,"abstract":"A recent breakthrough in Edmonds' problem showed that the noncommutative rank can be computed in deterministic polynomial time, and various algorithms for it were devised. However, only quite complicated algorithms are known for finding a so-called shrunk subspace, which acts as a dual certificate for the value of the noncommutative rank. In particular, the operator Sinkhorn algorithm, perhaps the simplest algorithm to compute the noncommutative rank with operator scaling, does not find a shrunk subspace. Finding a shrunk subspace plays a key role in applications, such as separation in the Brascamp-Lieb polytope, one-parameter subgroups in the null-cone membership problem, and primal-dual algorithms for matroid intersection and fractional matroid matching. In this paper, we provide a simple Sinkhorn-style algorithm to find the smallest shrunk subspace over the complex field in deterministic polynomial time. To this end, we introduce a generalization of the operator scaling problem, where the spectra of the marginals must be majorized by specified vectors. Then we design an efficient Sinkhorn-style algorithm for the generalized operator scaling problem. Applying this to the shrunk subspace problem, we show that a sufficiently long run of the algorithm also finds an approximate shrunk subspace close to the minimum exact shrunk subspace. Finally, we show that the approximate shrunk subspace can be rounded if it is sufficiently close. Along the way, we also provide a simple randomized algorithm to find the smallest shrunk subspace. As applications, we design a faster algorithm for fractional linear matroid matching and efficient weak membership and optimization algorithms for the rank-2 Brascamp-Lieb polytope.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"1 1","pages":"1655-1668"},"PeriodicalIF":0.0,"publicationDate":"2022-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87792562","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-16DOI: 10.1137/1.9781611977554.ch94
R. Ravi, Weizhong Zhang, Michael Zlatin
In the Steiner Tree Augmentation Problem (STAP), we are given a graph $G = (V,E)$, a set of terminals $R subseteq V$, and a Steiner tree $T$ spanning $R$. The edges $L := E setminus E(T)$ are called links and have non-negative costs. The goal is to augment $T$ by adding a minimum cost set of links, so that there are 2 edge-disjoint paths between each pair of vertices in $R$. This problem is a special case of the Survivable Network Design Problem, which can be approximated to within a factor of 2 using iterative rounding~cite{J2001}. We give the first polynomial time algorithm for STAP with approximation ratio better than 2. In particular, we achieve an approximation ratio of $(1.5 + varepsilon)$. To do this, we employ the Local Search approach of~cite{TZ2022} for the Tree Augmentation Problem and generalize their main decomposition theorem from links (of size two) to hyper-links. We also consider the Node-Weighted Steiner Tree Augmentation Problem (NW-STAP) in which the non-terminal nodes have non-negative costs. We seek a cheapest subset $S subseteq V setminus R$ so that $G[R cup S]$ is 2-edge-connected. Using a result of Nutov~cite{N2010}, there exists an $O(log |R|)$-approximation for this problem. We provide an $O(log^2 (|R|))$-approximation algorithm for NW-STAP using a greedy algorithm leveraging the spider decomposition of optimal solutions.
在斯坦纳树增强问题(STAP)中，我们给出了一个图$G = (V,E)$，一组终端$R subseteq V$，以及一个斯坦纳树$T$生成$R$。这些边$L := E setminus E(T)$被称为链接，它们的代价是非负的。我们的目标是通过添加一个最小代价的链接集来增强$T$，这样在$R$的每对顶点之间就有2条不相交的路径。这个问题是可生存网络设计问题的一个特殊情况，它可以使用迭代舍入cite{J2001}在因子2内近似。给出了第一个近似比大于2的多项式时间算法。特别地，我们得到了近似的比值$(1.5 + varepsilon)$。为了做到这一点，我们对树增强问题采用了cite{TZ2022}的局部搜索方法，并将它们的主要分解定理从链接(大小为2)推广到超链接。我们还考虑了节点加权斯坦纳树增强问题(NW-STAP)，其中非终端节点具有非负成本。我们寻找一个最便宜的子集$S subseteq V setminus R$，使得$G[R cup S]$是2边连接的。利用Nutov cite{N2010}的结果，这个问题存在一个$O(log |R|)$ -近似。我们使用贪婪算法利用最优解的蜘蛛分解为NW-STAP提供了$O(log^2 (|R|))$ -逼近算法。
{"title":"Approximation Algorithms for Steiner Tree Augmentation Problems","authors":"R. Ravi, Weizhong Zhang, Michael Zlatin","doi":"10.1137/1.9781611977554.ch94","DOIUrl":"https://doi.org/10.1137/1.9781611977554.ch94","url":null,"abstract":"In the Steiner Tree Augmentation Problem (STAP), we are given a graph $G = (V,E)$, a set of terminals $R subseteq V$, and a Steiner tree $T$ spanning $R$. The edges $L := E setminus E(T)$ are called links and have non-negative costs. The goal is to augment $T$ by adding a minimum cost set of links, so that there are 2 edge-disjoint paths between each pair of vertices in $R$. This problem is a special case of the Survivable Network Design Problem, which can be approximated to within a factor of 2 using iterative rounding~cite{J2001}. We give the first polynomial time algorithm for STAP with approximation ratio better than 2. In particular, we achieve an approximation ratio of $(1.5 + varepsilon)$. To do this, we employ the Local Search approach of~cite{TZ2022} for the Tree Augmentation Problem and generalize their main decomposition theorem from links (of size two) to hyper-links. We also consider the Node-Weighted Steiner Tree Augmentation Problem (NW-STAP) in which the non-terminal nodes have non-negative costs. We seek a cheapest subset $S subseteq V setminus R$ so that $G[R cup S]$ is 2-edge-connected. Using a result of Nutov~cite{N2010}, there exists an $O(log |R|)$-approximation for this problem. We provide an $O(log^2 (|R|))$-approximation algorithm for NW-STAP using a greedy algorithm leveraging the spider decomposition of optimal solutions.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"1 1","pages":"2429-2448"},"PeriodicalIF":0.0,"publicationDate":"2022-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79901257","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-16DOI: 10.48550/arXiv.2207.07974
Binghui Peng, Fred Zhang
We provide the first sub-linear space and sub-linear regret algorithm for online learning with expert advice (against an oblivious adversary), addressing an open question raised recently by Srinivas, Woodruff, Xu and Zhou (STOC 2022). We also demonstrate a separation between oblivious and (strong) adaptive adversaries by proving a linear memory lower bound of any sub-linear regret algorithm against an adaptive adversary. Our algorithm is based on a novel pool selection procedure that bypasses the traditional wisdom of leader selection for online learning, and a generic reduction that transforms any weakly sub-linear regret $o(T)$ algorithm to $T^{1-alpha}$ regret algorithm, which may be of independent interest. Our lower bound utilizes the connection of no-regret learning and equilibrium computation in zero-sum games, leading to a proof of a strong lower bound against an adaptive adversary.
{"title":"Online Prediction in Sub-linear Space","authors":"Binghui Peng, Fred Zhang","doi":"10.48550/arXiv.2207.07974","DOIUrl":"https://doi.org/10.48550/arXiv.2207.07974","url":null,"abstract":"We provide the first sub-linear space and sub-linear regret algorithm for online learning with expert advice (against an oblivious adversary), addressing an open question raised recently by Srinivas, Woodruff, Xu and Zhou (STOC 2022). We also demonstrate a separation between oblivious and (strong) adaptive adversaries by proving a linear memory lower bound of any sub-linear regret algorithm against an adaptive adversary. Our algorithm is based on a novel pool selection procedure that bypasses the traditional wisdom of leader selection for online learning, and a generic reduction that transforms any weakly sub-linear regret $o(T)$ algorithm to $T^{1-alpha}$ regret algorithm, which may be of independent interest. Our lower bound utilizes the connection of no-regret learning and equilibrium computation in zero-sum games, leading to a proof of a strong lower bound against an adaptive adversary.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"4 1","pages":"1611-1634"},"PeriodicalIF":0.0,"publicationDate":"2022-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89044478","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-16DOI: 10.48550/arXiv.2207.07949
C. Grunau, Ahmet Alper Ozudougru, Václav Rozhoň, Jakub Tvetek
The famous $k$-means++ algorithm of Arthur and Vassilvitskii [SODA 2007] is the most popular way of solving the $k$-means problem in practice. The algorithm is very simple: it samples the first center uniformly at random and each of the following $k-1$ centers is then always sampled proportional to its squared distance to the closest center so far. Afterward, Lloyd's iterative algorithm is run. The $k$-means++ algorithm is known to return a $Theta(log k)$ approximate solution in expectation. In their seminal work, Arthur and Vassilvitskii [SODA 2007] asked about the guarantees for its following emph{greedy} variant: in every step, we sample $ell$ candidate centers instead of one and then pick the one that minimizes the new cost. This is also how $k$-means++ is implemented in e.g. the popular Scikit-learn library [Pedregosa et al.; JMLR 2011]. We present nearly matching lower and upper bounds for the greedy $k$-means++: We prove that it is an $O(ell^3 log^3 k)$-approximation algorithm. On the other hand, we prove a lower bound of $Omega(ell^3 log^3 k / log^2(elllog k))$. Previously, only an $Omega(ell log k)$ lower bound was known [Bhattacharya, Eube, R"oglin, Schmidt; ESA 2020] and there was no known upper bound.
{"title":"A Nearly Tight Analysis of Greedy k-means++","authors":"C. Grunau, Ahmet Alper Ozudougru, Václav Rozhoň, Jakub Tvetek","doi":"10.48550/arXiv.2207.07949","DOIUrl":"https://doi.org/10.48550/arXiv.2207.07949","url":null,"abstract":"The famous $k$-means++ algorithm of Arthur and Vassilvitskii [SODA 2007] is the most popular way of solving the $k$-means problem in practice. The algorithm is very simple: it samples the first center uniformly at random and each of the following $k-1$ centers is then always sampled proportional to its squared distance to the closest center so far. Afterward, Lloyd's iterative algorithm is run. The $k$-means++ algorithm is known to return a $Theta(log k)$ approximate solution in expectation. In their seminal work, Arthur and Vassilvitskii [SODA 2007] asked about the guarantees for its following emph{greedy} variant: in every step, we sample $ell$ candidate centers instead of one and then pick the one that minimizes the new cost. This is also how $k$-means++ is implemented in e.g. the popular Scikit-learn library [Pedregosa et al.; JMLR 2011]. We present nearly matching lower and upper bounds for the greedy $k$-means++: We prove that it is an $O(ell^3 log^3 k)$-approximation algorithm. On the other hand, we prove a lower bound of $Omega(ell^3 log^3 k / log^2(elllog k))$. Previously, only an $Omega(ell log k)$ lower bound was known [Bhattacharya, Eube, R\"oglin, Schmidt; ESA 2020] and there was no known upper bound.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"84 1","pages":"1012-1070"},"PeriodicalIF":0.0,"publicationDate":"2022-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77136275","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-16DOI: 10.48550/arXiv.2207.07809
Siu-Wing Cheng, Haoqiang Huang
We present new approximation results on curve simplification and clustering under Fr'echet distance. Let $T = {tau_i : i in [n] }$ be polygonal curves in $R^d$ of $m$ vertices each. Let $l$ be any integer from $[m]$. We study a generalized curve simplification problem: given error bounds $delta_i>0$ for $i in [n]$, find a curve $sigma$ of at most $l$ vertices such that $d_F(sigma,tau_i) le delta_i$ for $i in [n]$. We present an algorithm that returns a null output or a curve $sigma$ of at most $l$ vertices such that $d_F(sigma,tau_i) le delta_i + epsilondelta_{max}$ for $i in [n]$, where $delta_{max} = max_{i in [n]} delta_i$. If the output is null, there is no curve of at most $l$ vertices within a Fr'echet distance of $delta_i$ from $tau_i$ for $i in [n]$. The running time is $tilde{O}bigl(n^{O(l)} m^{O(l^2)} (dl/epsilon)^{O(dl)}bigr)$. This algorithm yields the first polynomial-time bicriteria approximation scheme to simplify a curve $tau$ to another curve $sigma$, where the vertices of $sigma$ can be anywhere in $R^d$, so that $d_F(sigma,tau) le (1+epsilon)delta$ and $|sigma| le (1+alpha) min{|c| : d_F(c,tau) le delta}$ for any given $delta>0$ and any fixed $alpha, epsilon in (0,1)$. The running time is $tilde{O}bigl(m^{O(1/alpha)} (d/(alphaepsilon))^{O(d/alpha)}bigr)$. By combining our technique with some previous results in the literature, we obtain an approximation algorithm for $(k,l)$-median clustering. Given $T$, it computes a set $Sigma$ of $k$ curves, each of $l$ vertices, such that $sum_{i in [n]} min_{sigma in Sigma} d_F(sigma,tau_i)$ is within a factor $1+epsilon$ of the optimum with probability at least $1-mu$ for any given $mu, epsilon in (0,1)$. The running time is $tilde{O}bigl(n m^{O(kl^2)} mu^{-O(kl)} (dkl/epsilon)^{O((dkl/epsilon)log(1/mu))}bigr)$.
我们给出了新的曲线简化和聚类的近似结果。让 $T = {tau_i : i in [n] }$ 在多边形曲线中 $R^d$ 的 $m$ 每个顶点。让 $l$ 是以下任意整数 $[m]$． 研究了给定误差界的广义曲线化简问题 $delta_i>0$ 为了 $i in [n]$，找到一条曲线 $sigma$ 最多的 $l$ 这样的顶点 $d_F(sigma,tau_i) le delta_i$ 为了 $i in [n]$． 我们提出了一种返回空输出或曲线的算法 $sigma$ 最多的 $l$ 这样的顶点 $d_F(sigma,tau_i) le delta_i + epsilondelta_{max}$ 为了 $i in [n]$，其中 $delta_{max} = max_{i in [n]} delta_i$． 如果输出为空，则没有最多为的曲线 $l$ 的距离内的顶点 $delta_i$ 从 $tau_i$ 为了 $i in [n]$． 运行时间为 $tilde{O}bigl(n^{O(l)} m^{O(l^2)} (dl/epsilon)^{O(dl)}bigr)$． 该算法产生了简化曲线的第一个多项式时间双准则近似方案 $tau$ 到另一条曲线 $sigma$的顶点 $sigma$ 可以在任何地方 $R^d$，所以 $d_F(sigma,tau) le (1+epsilon)delta$ 和 $|sigma| le (1+alpha) min{|c| : d_F(c,tau) le delta}$ 对于任何给定的 $delta>0$ 任何固定的 $alpha, epsilon in (0,1)$． 运行时间为 $tilde{O}bigl(m^{O(1/alpha)} (d/(alphaepsilon))^{O(d/alpha)}bigr)$． 通过将我们的技术与文献中一些先前的结果相结合，我们得到了一个近似算法 $(k,l)$-中位数聚类。给定 $T$，它计算一个集合 $Sigma$ 的 $k$ 曲线，每一个 $l$ 顶点，这样 $sum_{i in [n]} min_{sigma in Sigma} d_F(sigma,tau_i)$ 在一个因素之内 $1+epsilon$ 最优值的概率 $1-mu$ 对于任何给定的 $mu, epsilon in (0,1)$． 运行时间为 $tilde{O}bigl(n m^{O(kl^2)} mu^{-O(kl)} (dkl/epsilon)^{O((dkl/epsilon)log(1/mu))}bigr)$．
{"title":"Curve Simplification and Clustering under Fréchet Distance","authors":"Siu-Wing Cheng, Haoqiang Huang","doi":"10.48550/arXiv.2207.07809","DOIUrl":"https://doi.org/10.48550/arXiv.2207.07809","url":null,"abstract":"We present new approximation results on curve simplification and clustering under Fr'echet distance. Let $T = {tau_i : i in [n] }$ be polygonal curves in $R^d$ of $m$ vertices each. Let $l$ be any integer from $[m]$. We study a generalized curve simplification problem: given error bounds $delta_i>0$ for $i in [n]$, find a curve $sigma$ of at most $l$ vertices such that $d_F(sigma,tau_i) le delta_i$ for $i in [n]$. We present an algorithm that returns a null output or a curve $sigma$ of at most $l$ vertices such that $d_F(sigma,tau_i) le delta_i + epsilondelta_{max}$ for $i in [n]$, where $delta_{max} = max_{i in [n]} delta_i$. If the output is null, there is no curve of at most $l$ vertices within a Fr'echet distance of $delta_i$ from $tau_i$ for $i in [n]$. The running time is $tilde{O}bigl(n^{O(l)} m^{O(l^2)} (dl/epsilon)^{O(dl)}bigr)$. This algorithm yields the first polynomial-time bicriteria approximation scheme to simplify a curve $tau$ to another curve $sigma$, where the vertices of $sigma$ can be anywhere in $R^d$, so that $d_F(sigma,tau) le (1+epsilon)delta$ and $|sigma| le (1+alpha) min{|c| : d_F(c,tau) le delta}$ for any given $delta>0$ and any fixed $alpha, epsilon in (0,1)$. The running time is $tilde{O}bigl(m^{O(1/alpha)} (d/(alphaepsilon))^{O(d/alpha)}bigr)$. By combining our technique with some previous results in the literature, we obtain an approximation algorithm for $(k,l)$-median clustering. Given $T$, it computes a set $Sigma$ of $k$ curves, each of $l$ vertices, such that $sum_{i in [n]} min_{sigma in Sigma} d_F(sigma,tau_i)$ is within a factor $1+epsilon$ of the optimum with probability at least $1-mu$ for any given $mu, epsilon in (0,1)$. The running time is $tilde{O}bigl(n m^{O(kl^2)} mu^{-O(kl)} (dkl/epsilon)^{O((dkl/epsilon)log(1/mu))}bigr)$.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"24 1","pages":"1414-1432"},"PeriodicalIF":0.0,"publicationDate":"2022-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78221470","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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