We have four papers this month — three on sublinear-time graph algorithms and one on distribution testing!

**Beating Greedy Matching in Sublinear Time**, by Soheil Behnezhad, Mohammad Roghani, Aviad Rubinstein, and Amin Saberi (arXiv). Designing sublinear-time algorithms to estimate the size of maximum matching in a graph is a well-studied problem. This paper gives the first \(\frac{1}{2} + \Omega(1)\) approximation algorithm that runs in time sublinear in the size of the input graph. Specifically, given a graph on \(n\) vertices and maximum degree \(\Delta\) in the adjacency list model, and a parameter \(\epsilon >0\), the algorithm runs in time \(\tilde{O}(n + \Delta^{1+\epsilon})\) and produces a \(\frac{1}{2} + f(\epsilon)\) approximation to the maximum matching for some function \(f\). It must be noted that a seminal work of Yoshida, Yamamoto and Ito (STOC, 2009) also gives a better than \(\frac{1}{2}\) approximation sublinear-time algorithm for the same problem. However, the result of Yoshida et al. requires assumptions on the maximum degree of the input graph. An additional point worth mentioning is that the authors do not believe that their techniques will yield an approximation guarantee better than \(0.51\), i.e., \(f(\epsilon) < 0.01\) for all \(\epsilon\).

**Sublinear-Time Clustering Oracle for Signed Graphs**, by Stefan Neumann and Pan Peng (arXiv). Consider a large *signed graph* on \(n\) vertices where vertices represent users of a social network and signed edges (+/-) denote the type of interactions (friendly or hostile) between users. Assume that the vertices of the social network can be partitioned into \(O(\log n)\) large clusters, where each cluster has a sparse cut with the rest of the graph. Further, each cluster is a minimal set (w.r.t. inclusion) that can be partitioned into roughly equal-sized opposing sub-communities, where a sub-community opposes another sub-community if most of the edges going across are negatively signed and most of the edges within the sub-communities are positively signed. This work provides a local oracle that, given probe access to a signed graph with such a hidden cluster structure, answers queries of the form “What cluster does vertex \(v\) belong to?” in time \(\tilde{O}(\sqrt{n} \cdot \text{poly}(1/\epsilon))\) per query. This result is a generalization of the same problem studied for unsigned graphs (Peng, 2020). The authors additionally show that their method works well in practice using both synthetic and real-world datasets. They also provide the first public real-world datasets of large signed graphs with a small number of large ground-truth communities having this property.

**Sublinear Algorithms for Hierarchical Clustering**, by Arpit Agarwal, Sanjeev Khanna, Huan Li, and Prathamesh Patil (arXiv). Consider a weighted graph \(G = (V,E,w)\), where the set \(V\) of vertices denotes datapoints and the weight \(w(e) > 0\) of edge \(e \in E\) denotes the similarity between the endpoints of \(e\). A hierarchical clustering of \(V\) is a tree \(T\) whose root is the set \(V\) and leaves are the singleton sets corresponding to individual vertices. An internal node of the tree corresponds to a cluster containing all the leaf vertices that are descendants of that node. A hierarchical clustering tree provides us with a scheme to cluster datapoints at multiple levels of granularity. The cost of a hierarchical clustering tree is \(\sum_{(u,v) \in E} |T_{u,v}| \cdot w(u,v)\), where \(T_{u,v}\) denotes the lowest common ancestor of the leaves \(u\) and \(v\). In this paper, the authors present sublinear algorithms for determining a hierarchical clustering tree with the minimum cost. In the query model with degree queries and neighbor queries to the graph, they give an algorithm that outputs an \(\tilde{O}(1)\)-approximate hierarchical clustering and makes \(\tilde{O}(n^{4-2\gamma})\) queries, when the number of edges \(m = \Theta(n^{\gamma})\) for \(1.5 \geq \gamma > 4/3\). When the input graph is sparse, i.e., \(\gamma \leq 4/3\), the algorithm makes \(\tilde{O}(\max\{n, m\})\) queries, and when the graph is dense, i.e., \(\gamma >1.5\), the algorithm makes \(\tilde{O}(n)\) queries. They complement their upper bounds with nearly tight lower bounds. In order to obtain their upper bounds, they design a sublinear-time algorithm for the problem of obtaining a *weak* cut sparsifier that approximates cuts sizes upto an additive term in addition to the usual multiplicative factor. They also design sublinear algorithms for hierarchical clustering in the MPC and streaming models of computation.

**Sharp Constants in Uniformity Testing via the Huber Statistic**, by Shivam Gupta and Eric Price (arXiv). This paper revisits the fundamental problem of uniformity testing — i.e., to decide whether an unknown distribution over \(n\) elements is uniform or \(\epsilon\)-far from uniform. This problem is known to be solvable optimally with probability at least \(1 – \delta\) using \(s = \Theta\left(\frac{\sqrt{n \log (1/\delta)}}{\epsilon^2} + \frac{\log (1/\delta)}{\epsilon^2}\right)\) independent samples from the unknown distribution. Multiple testers are known for the problem and they all compute a statistic of the form \(\sum_{i \in [n]} f(s_i)\), where \(s_i\) for \(i \in [n]\) and \(f\) is some function and make their decision based on whether or not the value of the statistic is above or below a threshold. For instance, the earliest known uniformity tester (Batu, Fortnow, Rubinfeld, Smith and White 2000; Goldreich and Ron 2011), also called the *collisions tester*, uses \(f(k) = \frac{k(k-1)}{2}\). The current paper proposes a new tester based on the Huber loss. For \(\beta > 0\), let \(h_\beta(x) := \min\{x^2, 2\beta x – \beta^2\}\). The statistic that the authors use in their test is defined by the function \(f(k) := k – s/n\), where \(s\) is the number of samples and \(n\) is the support size of the distribution. The authors show that their tester is better than all previously known testers as they achieve the best constants in the sample complexity.

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