Pub Date : 2023-01-01DOI: 10.1137/1.9781611977554.ch119
Peter Gartland, D. Lokshtanov
{"title":"Graph Classes with Few Minimal Separators. I. Finite Forbidden Induced Subgraphs","authors":"Peter Gartland, D. Lokshtanov","doi":"10.1137/1.9781611977554.ch119","DOIUrl":"https://doi.org/10.1137/1.9781611977554.ch119","url":null,"abstract":"","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"69 1","pages":"3063-3097"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85335401","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 : 2023-01-01DOI: 10.1137/1.9781611977554.ch59
Ngoc Mai Le, S. Umboh, Ningyuan Xie
{"title":"The Power of Clairvoyance for Multi-Level Aggregation and Set Cover with Delay","authors":"Ngoc Mai Le, S. Umboh, Ningyuan Xie","doi":"10.1137/1.9781611977554.ch59","DOIUrl":"https://doi.org/10.1137/1.9781611977554.ch59","url":null,"abstract":"","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"50 1","pages":"1594-1610"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90986359","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 : 2023-01-01DOI: 10.1137/1.9781611977554.ch188
M. Koucký, M. Saks
{"title":"Simple, deterministic, fast (but weak) approximations to edit distance and Dyck edit distance","authors":"M. Koucký, M. Saks","doi":"10.1137/1.9781611977554.ch188","DOIUrl":"https://doi.org/10.1137/1.9781611977554.ch188","url":null,"abstract":"","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"26 1","pages":"5203-5219"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74566470","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 : 2023-01-01DOI: 10.1137/1.9781611977554.ch23
Corwin Sinnamon, R. Tarjan
{"title":"A Nearly-Tight Analysis of Multipass Pairing Heaps","authors":"Corwin Sinnamon, R. Tarjan","doi":"10.1137/1.9781611977554.ch23","DOIUrl":"https://doi.org/10.1137/1.9781611977554.ch23","url":null,"abstract":"","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"117 ","pages":"535-548"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72544811","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 : 2023-01-01DOI: 10.1137/1.9781611977554.ch117
J. Cardinal, H. P. Hoang, Arturo I. Merino, Torsten Mütze
In 1993, Savage, Squire, and West described an inductive construction for generating every acyclic orientation of a chordal graph exactly once, flipping one arc at a time. We provide two generalizations of this result. Firstly, we describe Gray codes for acyclic orientations of hypergraphs that satisfy a simple ordering condition, which generalizes the notion of perfect elimination order of graphs. This unifies the Savage-Squire-West construction with a recent algorithm for generating elimination trees of chordal graphs (SODA 2022). Secondly, we consider quotients of lattices of acyclic orientations of chordal graphs, and we provide a Gray code for them, addressing a question raised by Pilaud (FPSAC 2022). This also generalizes a recent algorithm for generating lattice congruences of the weak order on the symmetric group (SODA 2020). Our algorithms are derived from the Hartung-Hoang-M¨utze-Williams combinatorial generation framework, and they yield simple algorithms for computing Hamilton paths and cycles on large classes of polytopes, including chordal nestohedra and quotientopes. In particular, we derive an efficient implementation of the Savage-Squire-West construction. Along the way, we give an overview of old and recent results about the polyhedral and order-theoretic aspects of acyclic orientations of graphs and hypergraphs.
{"title":"Zigzagging through acyclic orientations of chordal graphs and hypergraphs","authors":"J. Cardinal, H. P. Hoang, Arturo I. Merino, Torsten Mütze","doi":"10.1137/1.9781611977554.ch117","DOIUrl":"https://doi.org/10.1137/1.9781611977554.ch117","url":null,"abstract":"In 1993, Savage, Squire, and West described an inductive construction for generating every acyclic orientation of a chordal graph exactly once, flipping one arc at a time. We provide two generalizations of this result. Firstly, we describe Gray codes for acyclic orientations of hypergraphs that satisfy a simple ordering condition, which generalizes the notion of perfect elimination order of graphs. This unifies the Savage-Squire-West construction with a recent algorithm for generating elimination trees of chordal graphs (SODA 2022). Secondly, we consider quotients of lattices of acyclic orientations of chordal graphs, and we provide a Gray code for them, addressing a question raised by Pilaud (FPSAC 2022). This also generalizes a recent algorithm for generating lattice congruences of the weak order on the symmetric group (SODA 2020). Our algorithms are derived from the Hartung-Hoang-M¨utze-Williams combinatorial generation framework, and they yield simple algorithms for computing Hamilton paths and cycles on large classes of polytopes, including chordal nestohedra and quotientopes. In particular, we derive an efficient implementation of the Savage-Squire-West construction. Along the way, we give an overview of old and recent results about the polyhedral and order-theoretic aspects of acyclic orientations of graphs and hypergraphs.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"28 1","pages":"3029-3042"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82002250","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 : 2023-01-01DOI: 10.1137/1.9781611977554.ch30
Eli Ben-Sasson, Dan Carmon, Swastik Kopparty, D. Levit
{"title":"Elliptic Curve Fast Fourier Transform (ECFFT) Part I: Low-degree Extension in Time O(n log n) over all Finite Fields","authors":"Eli Ben-Sasson, Dan Carmon, Swastik Kopparty, D. Levit","doi":"10.1137/1.9781611977554.ch30","DOIUrl":"https://doi.org/10.1137/1.9781611977554.ch30","url":null,"abstract":"","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"15 1","pages":"700-737"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89427944","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 : 2022-12-19DOI: 10.48550/arXiv.2212.09348
Archontia C. Giannopoulou, D. Thilikos, S. Wiederrecht
noindent By a seminal result of Valiant, computing the permanent of $(0,1)$-matrices is, in general, $#mathsf{P}$-hard. In 1913 P'olya asked for which $(0,1)$-matrices $A$ it is possible to change some signs such that the permanent of $A$ equals the determinant of the resulting matrix. In 1975, Little showed these matrices to be exactly the biadjacency matrices of bipartite graphs excluding $K_{3,3}$ as a {matching minor}. This was turned into a polynomial time algorithm by McCuaig, Robertson, Seymour, and Thomas in 1999. However, the relation between the exclusion of some matching minor in a bipartite graph and the tractability of the permanent extends beyond $K_{3,3}.$ Recently it was shown that the exclusion of any planar bipartite graph as a matching minor yields a class of bipartite graphs on which the {permanent} of the corresponding $(0,1)$-matrices can be computed efficiently. In this paper we unify the two results above into a single, more general result in the style of the celebrated structure theorem for single-crossing-minor-free graphs. We identify a class of bipartite graphs strictly generalising planar bipartite graphs and $K_{3,3}$ which includes infinitely many non-Pfaffian graphs. The exclusion of any member of this class as a matching minor yields a structure that allows for the efficient evaluation of the permanent. Moreover, we show that the evaluation of the permanent remains $#mathsf{P}$-hard on bipartite graphs which exclude $K_{5,5}$ as a matching minor. This establishes a first computational lower bound for the problem of counting perfect matchings on matching minor closed classes.
{"title":"Excluding Single-Crossing Matching Minors in Bipartite Graphs","authors":"Archontia C. Giannopoulou, D. Thilikos, S. Wiederrecht","doi":"10.48550/arXiv.2212.09348","DOIUrl":"https://doi.org/10.48550/arXiv.2212.09348","url":null,"abstract":"noindent By a seminal result of Valiant, computing the permanent of $(0,1)$-matrices is, in general, $#mathsf{P}$-hard. In 1913 P'olya asked for which $(0,1)$-matrices $A$ it is possible to change some signs such that the permanent of $A$ equals the determinant of the resulting matrix. In 1975, Little showed these matrices to be exactly the biadjacency matrices of bipartite graphs excluding $K_{3,3}$ as a {matching minor}. This was turned into a polynomial time algorithm by McCuaig, Robertson, Seymour, and Thomas in 1999. However, the relation between the exclusion of some matching minor in a bipartite graph and the tractability of the permanent extends beyond $K_{3,3}.$ Recently it was shown that the exclusion of any planar bipartite graph as a matching minor yields a class of bipartite graphs on which the {permanent} of the corresponding $(0,1)$-matrices can be computed efficiently. In this paper we unify the two results above into a single, more general result in the style of the celebrated structure theorem for single-crossing-minor-free graphs. We identify a class of bipartite graphs strictly generalising planar bipartite graphs and $K_{3,3}$ which includes infinitely many non-Pfaffian graphs. The exclusion of any member of this class as a matching minor yields a structure that allows for the efficient evaluation of the permanent. Moreover, we show that the evaluation of the permanent remains $#mathsf{P}$-hard on bipartite graphs which exclude $K_{5,5}$ as a matching minor. This establishes a first computational lower bound for the problem of counting perfect matchings on matching minor closed classes.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"17 1","pages":"2111-2121"},"PeriodicalIF":0.0,"publicationDate":"2022-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85603387","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 : 2022-12-06DOI: 10.48550/arXiv.2212.03016
Ashish Chiplunkar, Monika Henzinger, Sagar Sudhir Kale, Maximilian Vötsch
Motivated by fairness requirements in communication networks, we introduce a natural variant of the online paging problem, called textit{min-max} paging, where the objective is to minimize the maximum number of faults on any page. While the classical paging problem, whose objective is to minimize the total number of faults, admits $k$-competitive deterministic and $O(log k)$-competitive randomized algorithms, we show that min-max paging does not admit a $c(k)$-competitive algorithm for any function $c$. Specifically, we prove that the randomized competitive ratio of min-max paging is $Omega(log(n))$ and its deterministic competitive ratio is $Omega(klog(n)/log(k))$, where $n$ is the total number of pages ever requested. We design a fractional algorithm for paging with a more general objective -- minimize the value of an $n$-variate differentiable convex function applied to the vector of the number of faults on each page. This gives an $O(log(n)log(k))$-competitive fractional algorithm for min-max paging. We show how to round such a fractional algorithm with at most a $k$ factor loss in the competitive ratio, resulting in a deterministic $O(klog(n)log(k))$-competitive algorithm for min-max paging. This matches our lower bound modulo a $mathrm{poly}(log(k))$ factor. We also give a randomized rounding algorithm that results in a $O(log^2 n log k)$-competitive algorithm.
由于通信网络中的公平性要求，我们引入了在线分页问题的一个自然变体，称为textit{最小-}最大分页，其目标是最小化任何页面上的最大错误数。而经典的分页问题，其目标是最小化故障总数，承认$k$ -竞争确定性和$O(log k)$ -竞争随机算法，我们表明，最小-最大分页不承认$c(k)$ -竞争算法对于任何函数$c$。具体来说，我们证明了最小-最大分页的随机竞争比为$Omega(log(n))$，其确定性竞争比为$Omega(klog(n)/log(k))$，其中$n$为所请求的总页数。我们为分页设计了一个分数算法，它具有更一般的目标——最小化应用于每个页面上的故障数量向量的$n$变量可微凸函数的值。这为最小-最大分页提供了一个$O(log(n)log(k))$竞争分数算法。我们展示了如何在竞争比率中至多以$k$因子损失舍入这样一个分数算法，从而产生最小最大分页的确定性$O(klog(n)log(k))$竞争算法。这与对$mathrm{poly}(log(k))$因子取模的下界相匹配。我们还给出了一个随机四舍五入算法，结果是一个$O(log^2 n log k)$竞争算法。
{"title":"Online Min-Max Paging","authors":"Ashish Chiplunkar, Monika Henzinger, Sagar Sudhir Kale, Maximilian Vötsch","doi":"10.48550/arXiv.2212.03016","DOIUrl":"https://doi.org/10.48550/arXiv.2212.03016","url":null,"abstract":"Motivated by fairness requirements in communication networks, we introduce a natural variant of the online paging problem, called textit{min-max} paging, where the objective is to minimize the maximum number of faults on any page. While the classical paging problem, whose objective is to minimize the total number of faults, admits $k$-competitive deterministic and $O(log k)$-competitive randomized algorithms, we show that min-max paging does not admit a $c(k)$-competitive algorithm for any function $c$. Specifically, we prove that the randomized competitive ratio of min-max paging is $Omega(log(n))$ and its deterministic competitive ratio is $Omega(klog(n)/log(k))$, where $n$ is the total number of pages ever requested. We design a fractional algorithm for paging with a more general objective -- minimize the value of an $n$-variate differentiable convex function applied to the vector of the number of faults on each page. This gives an $O(log(n)log(k))$-competitive fractional algorithm for min-max paging. We show how to round such a fractional algorithm with at most a $k$ factor loss in the competitive ratio, resulting in a deterministic $O(klog(n)log(k))$-competitive algorithm for min-max paging. This matches our lower bound modulo a $mathrm{poly}(log(k))$ factor. We also give a randomized rounding algorithm that results in a $O(log^2 n log k)$-competitive algorithm.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"11 1","pages":"1545-1565"},"PeriodicalIF":0.0,"publicationDate":"2022-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82199560","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 : 2022-11-29DOI: 10.48550/arXiv.2211.15945
Ce Jin, Jakob Nogler
Longest Common Substring (LCS) is an important text processing problem, which has recently been investigated in the quantum query model. The decisional version of this problem, LCS with threshold $d$, asks whether two length-$n$ input strings have a common substring of length $d$. The two extreme cases, $d=1$ and $d=n$, correspond respectively to Element Distinctness and Unstructured Search, two fundamental problems in quantum query complexity. However, the intermediate case $1ll dll n$ was not fully understood. We show that the complexity of LCS with threshold $d$ smoothly interpolates between the two extreme cases up to $n^{o(1)}$ factors: LCS with threshold $d$ has a quantum algorithm in $n^{2/3+o(1)}/d^{1/6}$ query complexity and time complexity, and requires at least $Omega(n^{2/3}/d^{1/6})$ quantum query complexity. Our result improves upon previous upper bounds $tilde O(min {n/d^{1/2}, n^{2/3}})$ (Le Gall and Seddighin ITCS 2022, Akmal and Jin SODA 2022), and answers an open question of Akmal and Jin. Our main technical contribution is a quantum speed-up of the powerful String Synchronizing Set technique introduced by Kempa and Kociumaka (STOC 2019). It consistently samples $n/tau^{1-o(1)}$ synchronizing positions in the string depending on their length-$Theta(tau)$ contexts, and each synchronizing position can be reported by a quantum algorithm in $tilde O(tau^{1/2+o(1)})$ time. As another application of our quantum string synchronizing set, we study the $k$-mismatch Matching problem, which asks if the pattern has an occurrence in the text with at most $k$ Hamming mismatches. Using a structural result of Charalampopoulos, Kociumaka, and Wellnitz (FOCS 2020), we obtain a quantum algorithm for $k$-mismatch matching with $k^{3/4} n^{1/2+o(1)}$ query complexity and $tilde O(kn^{1/2})$ time complexity.
{"title":"Quantum Speed-ups for String Synchronizing Sets, Longest Common Substring, and k-mismatch Matching","authors":"Ce Jin, Jakob Nogler","doi":"10.48550/arXiv.2211.15945","DOIUrl":"https://doi.org/10.48550/arXiv.2211.15945","url":null,"abstract":"Longest Common Substring (LCS) is an important text processing problem, which has recently been investigated in the quantum query model. The decisional version of this problem, LCS with threshold $d$, asks whether two length-$n$ input strings have a common substring of length $d$. The two extreme cases, $d=1$ and $d=n$, correspond respectively to Element Distinctness and Unstructured Search, two fundamental problems in quantum query complexity. However, the intermediate case $1ll dll n$ was not fully understood. We show that the complexity of LCS with threshold $d$ smoothly interpolates between the two extreme cases up to $n^{o(1)}$ factors: LCS with threshold $d$ has a quantum algorithm in $n^{2/3+o(1)}/d^{1/6}$ query complexity and time complexity, and requires at least $Omega(n^{2/3}/d^{1/6})$ quantum query complexity. Our result improves upon previous upper bounds $tilde O(min {n/d^{1/2}, n^{2/3}})$ (Le Gall and Seddighin ITCS 2022, Akmal and Jin SODA 2022), and answers an open question of Akmal and Jin. Our main technical contribution is a quantum speed-up of the powerful String Synchronizing Set technique introduced by Kempa and Kociumaka (STOC 2019). It consistently samples $n/tau^{1-o(1)}$ synchronizing positions in the string depending on their length-$Theta(tau)$ contexts, and each synchronizing position can be reported by a quantum algorithm in $tilde O(tau^{1/2+o(1)})$ time. As another application of our quantum string synchronizing set, we study the $k$-mismatch Matching problem, which asks if the pattern has an occurrence in the text with at most $k$ Hamming mismatches. Using a structural result of Charalampopoulos, Kociumaka, and Wellnitz (FOCS 2020), we obtain a quantum algorithm for $k$-mismatch matching with $k^{3/4} n^{1/2+o(1)}$ query complexity and $tilde O(kn^{1/2})$ time complexity.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"51 1","pages":"5090-5121"},"PeriodicalIF":0.0,"publicationDate":"2022-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74677549","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 : 2022-11-21DOI: 10.48550/arXiv.2211.11328
Michael Kapralov, Hannah Lawrence, M. Makarov, Cameron Musco, K. Sheth
We present a sublinear query algorithm for outputting a near-optimal low-rank approximation to any positive semidefinite Toeplitz matrix $T in mathbb{R}^{d times d}$. In particular, for any integer rank $k leq d$ and $epsilon,delta>0$, our algorithm makes $tilde{O} left (k^2 cdot log(1/delta) cdot text{poly}(1/epsilon) right )$ queries to the entries of $T$ and outputs a rank $tilde{O} left (k cdot log(1/delta)/epsilonright )$ matrix $tilde{T} in mathbb{R}^{d times d}$ such that $| T - tilde{T}|_F leq (1+epsilon) cdot |T-T_k|_F + delta |T|_F$. Here, $|cdot|_F$ is the Frobenius norm and $T_k$ is the optimal rank-$k$ approximation to $T$, given by projection onto its top $k$ eigenvectors. $tilde{O}(cdot)$ hides $text{polylog}(d) $ factors. Our algorithm is emph{structure-preserving}, in that the approximation $tilde{T}$ is also Toeplitz. A key technical contribution is a proof that any positive semidefinite Toeplitz matrix in fact has a near-optimal low-rank approximation which is itself Toeplitz. Surprisingly, this basic existence result was not previously known. Building on this result, along with the well-established off-grid Fourier structure of Toeplitz matrices [Cybenko'82], we show that Toeplitz $tilde{T}$ with near optimal error can be recovered with a small number of random queries via a leverage-score-based off-grid sparse Fourier sampling scheme.
我们提出了一种次线性查询算法，用于输出任何正半定Toeplitz矩阵的近最优低秩逼近$T in mathbb{R}^{d times d}$。特别是，对于任何整数秩$k leq d$和$epsilon,delta>0$，我们的算法对$T$的条目进行$tilde{O} left (k^2 cdot log(1/delta) cdot text{poly}(1/epsilon) right )$查询，并输出一个秩$tilde{O} left (k cdot log(1/delta)/epsilonright )$矩阵$tilde{T} in mathbb{R}^{d times d}$，使得$| T - tilde{T}|_F leq (1+epsilon) cdot |T-T_k|_F + delta |T|_F$。这里，$|cdot|_F$是Frobenius范数，$T_k$是$T$的最优秩- $k$近似值，通过在其顶部$k$特征向量上的投影给出。$tilde{O}(cdot)$隐藏了$text{polylog}(d) $因素。我们的算法emph{是保持结构}的，因为近似$tilde{T}$也是Toeplitz。一个关键的技术贡献是证明了任何正半定Toeplitz矩阵实际上都有一个接近最优的低秩近似，它本身就是Toeplitz。令人惊讶的是，这个基本的存在性结果以前并不为人所知。在此结果的基础上，结合已建立的Toeplitz矩阵的离网傅立叶结构[Cybenko'82]，我们表明，通过基于杠杆分数的离网稀疏傅立叶采样方案，可以使用少量随机查询恢复具有接近最优误差的Toeplitz $tilde{T}$。
{"title":"Toeplitz Low-Rank Approximation with Sublinear Query Complexity","authors":"Michael Kapralov, Hannah Lawrence, M. Makarov, Cameron Musco, K. Sheth","doi":"10.48550/arXiv.2211.11328","DOIUrl":"https://doi.org/10.48550/arXiv.2211.11328","url":null,"abstract":"We present a sublinear query algorithm for outputting a near-optimal low-rank approximation to any positive semidefinite Toeplitz matrix $T in mathbb{R}^{d times d}$. In particular, for any integer rank $k leq d$ and $epsilon,delta>0$, our algorithm makes $tilde{O} left (k^2 cdot log(1/delta) cdot text{poly}(1/epsilon) right )$ queries to the entries of $T$ and outputs a rank $tilde{O} left (k cdot log(1/delta)/epsilonright )$ matrix $tilde{T} in mathbb{R}^{d times d}$ such that $| T - tilde{T}|_F leq (1+epsilon) cdot |T-T_k|_F + delta |T|_F$. Here, $|cdot|_F$ is the Frobenius norm and $T_k$ is the optimal rank-$k$ approximation to $T$, given by projection onto its top $k$ eigenvectors. $tilde{O}(cdot)$ hides $text{polylog}(d) $ factors. Our algorithm is emph{structure-preserving}, in that the approximation $tilde{T}$ is also Toeplitz. A key technical contribution is a proof that any positive semidefinite Toeplitz matrix in fact has a near-optimal low-rank approximation which is itself Toeplitz. Surprisingly, this basic existence result was not previously known. Building on this result, along with the well-established off-grid Fourier structure of Toeplitz matrices [Cybenko'82], we show that Toeplitz $tilde{T}$ with near optimal error can be recovered with a small number of random queries via a leverage-score-based off-grid sparse Fourier sampling scheme.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"77 1","pages":"4127-4158"},"PeriodicalIF":0.0,"publicationDate":"2022-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86230444","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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