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1. All-Norm Load Balancing in Graph Streams via the Multiplicative Weights Update Method

   https://doi.org/10.4230/LIPIcs.ITCS.2023.7  

   Assadi, Sepehr; Bernstein, Aaron; Langley, Zachary (January 2023, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Tauman_Kalai, Yael (Ed.)

   In the weighted load balancing problem, the input is an n-vertex bipartite graph between a set of clients and a set of servers, and each client comes with some nonnegative real weight. The output is an assignment that maps each client to one of its adjacent servers, and the load of a server is then the sum of the weights of the clients assigned to it. The goal is to find an assignment that is well-balanced, typically captured by (approximately) minimizing either the 𝓁_∞- or 𝓁₂-norm of the server loads. Generalizing both of these objectives, the all-norm load balancing problem asks for an assignment that approximately minimizes all 𝓁_p-norm objectives for p ≥ 1, including p = ∞, simultaneously. Our main result is a deterministic O(log n)-pass O(1)-approximation semi-streaming algorithm for the all-norm load balancing problem. Prior to our work, only an O(log n)-pass O(log n)-approximation algorithm for the 𝓁_∞-norm objective was known in the semi-streaming setting. Our algorithm uses a novel application of the multiplicative weights update method to a mixed covering/packing convex program for the all-norm load balancing problem involving an infinite number of constraints. 

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2. Local Correlation Clustering with Asymmetric Classification Errors

   Jafarov, Jafar; Kalhan, Sanchit; Makarychev, Konstantin; Makarychev, Yury (January 2021, International Conference on Machine Learning)

   null (Ed.)

   In the Correlation Clustering problem, we are given a complete weighted graph G with its edges labeled as “similar" and “dissimilar" by a noisy binary classifier. For a clustering C of graph G, a similar edge is in disagreement with C, if its endpoints belong to distinct clusters; and a dissimilar edge is in disagreement with C if its endpoints belong to the same cluster. The disagreements vector is a vector indexed by the vertices of G such that the v-th coordinate of the disagreements vector equals the weight of all disagreeing edges incident on v. The goal is to produce a clustering that minimizes the ℓp norm of the disagreements vector for p≥1. We study the ℓ_p objective in Correlation Clustering under the following assumption: Every similar edge has weight in [αw,w] and every dissimilar edge has weight at least αw (where α ≤ 1 and w > 0 is a scaling parameter). We give an O((1/α)^{1/2−1/(2p)} log 1/α) approximation algorithm for this problem. Furthermore, we show an almost matching convex programming integrality gap. 

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3. Local Correlation Clustering with Asymmetric Classification Errors

   Jafarov, Jafar; Kalhan, Sanchit; Makarychev, Konstantin; Makarychev, Yury (July 2021, International Conference on Machine Learning)

   null (Ed.)

   In the Correlation Clustering problem, we are given a complete weighted graph G with its edges labeled as “similar" and “dissimilar" by a noisy binary classifier. For a clustering C of graph G, a similar edge is in disagreement with C, if its endpoints belong to distinct clusters; and a dissimilar edge is in disagreement with C if its endpoints belong to the same cluster. The disagreements vector, Agree, is a vector indexed by the vertices of G such that the v-th coordinate Disagre equals the weight of all disagreeing edges incident on v. The goal is to produce a clustering that minimizes the ℓp norm of the disagreements vector for p≥1. We study the ℓ_p objective in Correlation Clustering under the following assumption: Every similar edge has weight in [αw,w] and every dissimilar edge has weight at least αw (where α≤1 and w>0 is a scaling parameter). We give an O((1/α)^{1/2−1/2p}⋅log(1/α)) approximation algorithm for this problem. Furthermore, we show an almost matching convex programming integrality gap. 

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4. Local Correlation Clustering with Asymmetric Classification Errors

   Jafarov, Jafar; Kalhan, Sanchit; Makarychev, Konstantin; Makarychev, Yury (January 2021, Proceedings of Machine Learning Research)

   Meila, Marina; Zhang, Tong (Ed.)

   In the Correlation Clustering problem, we are given a complete weighted graph $$G$$ with its edges labeled as “similar" and “dissimilar" by a noisy binary classifier. For a clustering $$\mathcal{C}$$ of graph $$G$$, a similar edge is in disagreement with $$\mathcal{C}$$, if its endpoints belong to distinct clusters; and a dissimilar edge is in disagreement with $$\mathcal{C}$$ if its endpoints belong to the same cluster. The disagreements vector, $$\mathbf{disagree}$$, is a vector indexed by the vertices of $$G$$ such that the $$v$$-th coordinate $$\mathbf{disagree}_v$$ equals the weight of all disagreeing edges incident on $$v$$. The goal is to produce a clustering that minimizes the $$\ell_p$$ norm of the disagreements vector for $$p\geq 1$$. We study the $$\ell_p$$ objective in Correlation Clustering under the following assumption: Every similar edge has weight in $$[\alpha\mathbf{w},\mathbf{w}]$$ and every dissimilar edge has weight at least $$\alpha\mathbf{w}$$ (where $$\alpha \leq 1$$ and $$\mathbf{w}>0$$ is a scaling parameter). We give an $$O\left((\frac{1}{\alpha})^{\frac{1}{2}-\frac{1}{2p}}\cdot \log\frac{1}{\alpha}\right)$$ approximation algorithm for this problem. Furthermore, we show an almost matching convex programming integrality gap. 

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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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