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1. An Optimal Streaming Algorithm for Submodular Maximization with a Cardinality Constraint

   https://doi.org/10.1287/moor.2021.1224  

   Alaluf, Naor ; Ene, Alina ; Feldman, Moran ; Nguyen, Huy L. ; Suh, Andrew ( February 2022 , Mathematics of operations research)

   We study the problem of maximizing a non-monotone submodular function subject to a cardinality constraint in the streaming model. Our main contribution is a single-pass (semi-)streaming algorithm that uses roughly $O(k / \varepsilon^2)$ memory, where $k$ is the size constraint. At the end of the stream, our algorithm post-processes its data structure using any offline algorithm for submodular maximization, and obtains a solution whose approximation guarantee is $\frac{\alpha}{1+\alpha}-\varepsilon$, where $\alpha$ is the approximation of the offline algorithm. If we use an exact (exponential time) post-processing algorithm, this leads to $\frac{1}{2}-\varepsilon$ approximation (which is nearly optimal). If we post-process with the algorithm of \cite{buchbinder2019constrained}, that achieves the state-of-the-art offline approximation guarantee of $\alpha=0.385$, we obtain $0.2779$-approximation in polynomial time, improving over the previously best polynomial-time approximation of $0.1715$ due to \cite{feldman2018less}. It is also worth mentioning that our algorithm is combinatorial and deterministic, which is rare for an algorithm for non-monotone submodular maximization, and enjoys a fast update time of $O(\frac{\log k + \log (1/\alpha {\varepsilon^2})$ per element. 

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2. Streaming Algorithm for Monotone k-Submodular Maximization with Cardinality Constraints

   Ene, Alina ; Nguyen, Huy ( January 2022 , Proceedings of Machine Learning Research)

   Maximizing a monotone k-submodular function subject to cardinality constraints is a general model for several applications ranging from influence maximization with multiple products to sensor placement with multiple sensor types and online ad allocation. Due to the large problem scale in many applications and the online nature of ad allocation, a need arises for algorithms that process elements in a streaming fashion and possibly make online decisions. In this work, we develop a new streaming algorithm for maximizing a monotone k-submodular function subject to a per-coordinate cardinality constraint attaining an approximation guarantee close to the state of the art guarantee in the offline setting. Though not typical for streaming algorithms, our streaming algorithm also readily applies to the online setting with free disposal. Our algorithm is combinatorial and enjoys fast running time and small number of function evaluations. Furthermore, its guarantee improves as the cardinality constraints get larger, which is especially suited for the large scale applications. For the special case of maximizing a submodular function with large budgets, our combinatorial algorithm matches the guarantee of the state-of-the-art continuous algorithm, which requires significantly more time and function evaluations. 

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3. Algorithms for covering multiple submodular constraints and applications

   https://doi.org/10.1007/s10878-022-00874-x  

   Chekuri, Chandra ; Inamdar, Tanmay ; Quanrud, Kent ; Varadarajan, Kasturi ; Zhang, Zhao ( June 2022 , Journal of Combinatorial Optimization)

   We consider the problem of covering multiple submodular constraints. Given a finite ground setN, a weight function$$w: N \rightarrow \mathbb {R}_+$$$w:N\to R_{+}$,rmonotone submodular functions$$f_1,f_2,\ldots ,f_r$$$f_1,f_2,\dots ,f_r$overNand requirements$$k_1,k_2,\ldots ,k_r$$$k_1,k_2,\dots ,k_r$the goal is to find a minimum weight subset$$S \subseteq N$$$S\subseteq N$such that$$f_i(S) \ge k_i$$$f_i\left(S\right)\ge k_i$for$$1 \le i \le r$$$1\le i\le r$. We refer to this problem asMulti-Submod-Coverand it was recently considered by Har-Peled and Jones (Few cuts meet many point sets. CoRR.arxiv:abs1808.03260Har-Peled and Jones 2018) who were motivated by an application in geometry. Even with$$r=1$$$r=1$Multi-Submod-Covergeneralizes the well-known Submodular Set Cover problem (Submod-SC), and it can also be easily reduced toSubmod-SC. A simple greedy algorithm gives an$$O(\log (kr))$$$O(log(kr\left)\right)$approximation where$$k = \sum _i k_i$$$k={\sum }_i{k}_i$and this ratio cannot be improved in the general case. In this paper, motivated by several concrete applications, we consider two ways to improve upon the approximation given by the greedy algorithm. First, we give a bicriteria approximation algorithm forMulti-Submod-Coverthat covers each constraint to within a factor of$$(1-1/e-\varepsilon )$$$(1-1/e-\epsilon )$while incurring an approximation of$$O(\frac{1}{\epsilon }\log r)$$$O(\frac{1}{ϵ}logr)$in the cost. Second, we consider the special case when each$$f_i$$$f_i$is a obtained from a truncated coverage function and obtain an algorithm that generalizes previous work on partial set cover (Partial-SC), covering integer programs (CIPs) and multiple vertex cover constraints Bera et al. (Theoret Comput Sci 555:2–8 Bera et al. 2014). Both these algorithms are based on mathematical programming relaxations that avoid the limitations of the greedy algorithm. We demonstrate the implications of our algorithms and related ideas to several applications ranging from geometric covering problems to clustering with outliers. Our work highlights the utility of the high-level model and the lens of submodularity in addressing this class of covering problems.

    

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4. Adaptive Out-Orientations with Applications

   Chekuri, Chandra ; Christiansen, Aleksander Bjørn ; Holm, Jacob ; van der Hoog, Ivor ; Quanrud, Kent ; Rotenberg, Eva ; Schwiegelshohn, Chris. ( January 2024 , Proceedings of the 2024 ACM-SIAM Symposium on Discrete Algorithms, SODA 2024, Alexandria, VA, USA, January 7-10, 2024)

   Woodruff, David P. (Ed.)

   We give improved algorithms for maintaining edge-orientations of a fully-dynamic graph, such that the maximum out-degree is bounded. On one hand, we show how to orient the edges such that maximum out-degree is proportional to the arboricity $\alpha$ of the graph, in, either, an amortised update time of $O(\log^2 n \log \alpha)$, or a worst-case update time of $O(\log^3 n \log \alpha)$. On the other hand, motivated by applications including dynamic maximal matching, we obtain a different trade-off. Namely, the improved update time of either $O(\log n \log \alpha)$, amortised, or $O(\log ^2 n \log \alpha)$, worst-case, for the problem of maintaining an edge-orientation with at most $O(\alpha + \log n)$ out-edges per vertex. Finally, all of our algorithms naturally limit the recourse to be polylogarithmic in $n$ and $\alpha$. Our algorithms adapt to the current arboricity of the graph, and yield improvements over previous work: Firstly, we obtain deterministic algorithms for maintaining a $(1+\varepsilon)$ approximation of the maximum subgraph density, $\rho$, of the dynamic graph. Our algorithms have update times of $O(\varepsilon^{-6}\log^3 n \log \rho)$ worst-case, and $O(\varepsilon^{-4}\log^2 n \log \rho)$ amortised, respectively. We may output a subgraph $H$ of the input graph where its density is a $(1+\varepsilon)$ approximation of the maximum subgraph density in time linear in the size of the subgraph. These algorithms have improved update time compared to the $O(\varepsilon^{-6}\log ^4 n)$ algorithm by Sawlani and Wang from STOC 2020. Secondly, we obtain an $O(\varepsilon^{-6}\log^3 n \log \alpha)$ worst-case update time algorithm for maintaining a $(1~+~\varepsilon)\textnormal{OPT} + 2$ approximation of the optimal out-orientation of a graph with adaptive arboricity $\alpha$, improving the $O(\varepsilon^{-6}\alpha^2 \log^3 n)$ algorithm by Christiansen and Rotenberg from ICALP 2022. This yields the first worst-case polylogarithmic dynamic algorithm for decomposing into $O(\alpha)$ forests. Thirdly, we obtain arboricity-adaptive fully-dynamic deterministic algorithms for a variety of problems including maximal matching, $\Delta+1$ colouring, and matrix vector multiplication. All update times are worst-case $O(\alpha+\log^2n \log \alpha)$, where $\alpha$ is the current arboricity of the graph. For the maximal matching problem, the state-of-the-art deterministic algorithms by Kopelowitz, Krauthgamer, Porat, and Solomon from ICALP 2014 runs in time $O(\alpha^2 + \log^2 n)$, and by Neiman and Solomon from STOC 2013 runs in time $O(\sqrt{m})$. We give improved running times whenever the arboricity $\alpha \in \omega( \log n\sqrt{\log\log n})$. 

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5. Tight Bounds for ℓ 1 Oblivious Subspace Embeddings

   https://doi.org/10.1145/3477537  

   Wang, Ruosong ; Woodruff, David P. ( January 2022 , ACM Transactions on Algorithms)

   An \ell _p oblivious subspace embedding is a distribution over r \times n matrices \Pi such that for any fixed n \times d matrix A , \[ \Pr _{\Pi }[\textrm {for all }x, \ \Vert Ax\Vert _p \le \Vert \Pi Ax\Vert _p \le \kappa \Vert Ax\Vert _p] \ge 9/10,\] where r is the dimension of the embedding, \kappa is the distortion of the embedding, and for an n -dimensional vector y , \Vert y\Vert _p = (\sum _{i=1}^n |y_i|^p)^{1/p} is the \ell _p -norm. Another important property is the sparsity of \Pi , that is, the maximum number of non-zero entries per column, as this determines the running time of computing \Pi A . While for p = 2 there are nearly optimal tradeoffs in terms of the dimension, distortion, and sparsity, for the important case of 1 \le p \lt 2 , much less was known. In this article, we obtain nearly optimal tradeoffs for \ell _1 oblivious subspace embeddings, as well as new tradeoffs for 1 \lt p \lt 2 . Our main results are as follows: (1) We show for every 1 \le p \lt 2 , any oblivious subspace embedding with dimension r has distortion \[ \kappa = \Omega \left(\frac{1}{\left(\frac{1}{d}\right)^{1 / p} \log ^{2 / p}r + \left(\frac{r}{n}\right)^{1 / p - 1 / 2}}\right).\] When r = {\operatorname{poly}}(d) \ll n in applications, this gives a \kappa = \Omega (d^{1/p}\log ^{-2/p} d) lower bound, and shows the oblivious subspace embedding of Sohler and Woodruff (STOC, 2011) for p = 1 is optimal up to {\operatorname{poly}}(\log (d)) factors. (2) We give sparse oblivious subspace embeddings for every 1 \le p \lt 2 . Importantly, for p = 1 , we achieve r = O(d \log d) , \kappa = O(d \log d) and s = O(\log d) non-zero entries per column. The best previous construction with s \le {\operatorname{poly}}(\log d) is due to Woodruff and Zhang (COLT, 2013), giving \kappa = \Omega (d^2 {\operatorname{poly}}(\log d)) or \kappa = \Omega (d^{3/2} \sqrt {\log n} \cdot {\operatorname{poly}}(\log d)) and r \ge d \cdot {\operatorname{poly}}(\log d) ; in contrast our r = O(d \log d) and \kappa = O(d \log d) are optimal up to {\operatorname{poly}}(\log (d)) factors even for dense matrices. We also give (1) \ell _p oblivious subspace embeddings with an expected 1+\varepsilon number of non-zero entries per column for arbitrarily small \varepsilon \gt 0 , and (2) the first oblivious subspace embeddings for 1 \le p \lt 2 with O(1) -distortion and dimension independent of n . Oblivious subspace embeddings are crucial for distributed and streaming environments, as well as entrywise \ell _p low-rank approximation. Our results give improved algorithms for these applications. 

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