We study lower bounds for estimating the spectral density of the normalized adjacency matrix of a graph. Previously, Cohen-Steiner et al. [KDD 2018] proposed an algorithm for $\varepsilon$-approximate spectral density estimation in the Wasserstein-1 distance, using $2^{O(1/\varepsilon)}$ random walks initiated from uniformly random nodes in the graph. Later, Jin et al. [COLT 2023] established a nearly matching exponential lower bound for \emph{weighted} graphs, assuming the algorithm has access to samples from random walks started at random nodes. It was left open whether this lower bound could be extended to \emph{unweighted} graphs. In this paper, we answer this question in the affirmative by proving an exponential lower bound for unweighted graphs. Specifically, we show that no algorithm can compute an $\varepsilon$-approximation to the spectrum of a normalized graph adjacency matrix with constant success probability, even when given the full transcripts of $2^{\Omega(1/\varepsilon^{1/6})}$ random walks, each of length $2^{\Omega(1/\varepsilon^{1/6})}$, started from uniformly random nodes.

Effective Resistance (ER) is a fundamental tool in various graph learning tasks. In this paper, we address the problem of efficiently approximating ER on a graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ with $n$ vertices and $m$ edges. First, we focus on local online-computation algorithms for ER approximation, aiming to improve the dependency on the approximation error parameter $\epsilon$. Specifically, for a given vertex pair $(s,t)$, we propose a local algorithm with a time complexity of $\tilde{O}(\sqrt{d}/\epsilon)$ to compute an $\epsilon$-approximation of the $s,t$-ER value for expander graphs, where $d=\min \{d_s,d_t\}$. This improves upon the previous state-of-the-art, including an $\tilde{O}(1/\epsilon^2)$ time algorithm based on random walk sampling by Andoni et al. (ITCS’19) and Peng et al. (KDD’21). Our method achieves this improvement by combining deterministic search with random walk sampling to reduce variance. Second, we establish a lower bound for ER approximation on expander graphs. We prove that for any $\epsilon\in (0,1)$, there exist an expander graph and a vertex pair $(s,t)$ such that any local algorithm requires at least $\Omega(1/\epsilon)$ time to compute the $\epsilon$-approximation of the $s,t$-ER value. Finally, we extend our techniques to index-based algorithms for ER computation. We propose an algorithm with $\tilde{O}(\min \{m+n/\epsilon^{1.5},\sqrt{nm}/\epsilon\})$ processing time, $\tilde{O}(n/\epsilon)$ space complexity and $O(1)$ query complexity, which returns an $\epsilon$-approximation of the $s,t$-ER value for any $s,t\in \mathcal{V}$ for expander graphs. Our approach improves upon the state-of-the-art $\tilde{O}(m/\epsilon)$ processing time by Dwaraknath et al. (NeurIPS’24) and the $\tilde{O}(m+n/\epsilon^2)$ processing time by Li and Sachdeva (SODA’23).

Multi-object tracking (MOT) in sports scenes plays a critical role in gathering players statistics, supporting further applications, such as automatic tactical analysis. Yet existing MOT benchmarks cast little attention on this domain. In this work, we present a new large-scale multi-object tracking dataset in multiple sports scenes, coined as SportsMOT, where all players on the court are supposed to be tracked. It consists of 240 video sequences, over 150K frames (almost 15x MOT17) and over 1.6M bounding boxes (3x MOT17) collected from 3 sports categories, including basketball, volleyball and football. Our dataset is characterized with two key properties: 1) fast and variable-speed motion and 2) similar yet distinguishable appearance. We expect SportsMOT to encourage the MOT trackers to promote in both motion-based association and appearance-based association. We benchmark several state-of-the-art trackers and reveal the key challenge of SportsMOT lies in object association. To alleviate the issue, we further propose a new multi-object tracking framework, termed as MixSort, introducing a MixFormer-like structure as an auxiliary association model to prevailing tracking-by-detection trackers. By integrating the customized appearance-based association with the original motion-based association, MixSort achieves state-of-the-art performance on SportsMOT and MOT17. Based on MixSort, we give an in-depth analysis and provide some profound insights into SportsMOT.