Multi-Swap $k$-Means++

The $k$-means++ algorithm of Arthur and Vassilvitskii (SODA 2007) is often the practitioners' choice algorithm for optimizing the popular $k$-means clustering objective and is known to give an $O(\log k)$-approximation in expectation. To obtain higher quality solutions, Lattanzi and Sohler (ICML 2019) proposed augmenting $k$-means++ with $O(k \log \log k)$ local search steps obtained through the $k$-means++ sampling distribution to yield a $c$-approximation to the $k$-means clustering problem, where $c$ is a large absolute constant. Here we generalize and extend their local search algorithm by considering larger and more sophisticated local search neighborhoods hence allowing to swap multiple centers at the same time. Our algorithm achieves a $9 + \varepsilon$ approximation ratio, which is the best possible for local search. Importantly we show that our approach yields substantial practical improvements, we show significant quality improvements over the approach of Lattanzi and Sohler (ICML 2019) on several datasets.

An Optimal Algorithm for Finding Champions in Tournament Graphs

A tournament graph $T = \left(V, E \right)$ is an oriented complete graph, which can be used to model a round-robin tournament between $n$ players. In this paper, we address the problem of finding a champion of the tournament, also known as Copeland winner, which is a player that wins the highest number of matches. Solving this problem has important implications on several Information Retrieval applications, including Web search, conversational IR, machine translation, question answering, recommender systems, etc. Our goal is to solve the problem by minimizing the number of times we probe the adjacency matrix, i.e., the number of matches played. We prove that any deterministic/randomized algorithm finding a champion with constant success probability requires $\Omega(\ell n)$ comparisons, where $\ell$ is the number of matches lost by the champion. We then present an optimal deterministic algorithm matching this lower bound without knowing $\ell$ and we extend our analysis to three strictly related problems. Lastly, we conduct a comprehensive experimental assessment of the proposed algorithms to speed up a state-of-the-art solution for ranking on public data. Results show that our proposals speed up the retrieval of the champion up to $13\times$ in this scenario.