The classical center based clustering problems such as $k$-means/median/center assume that the optimal clusters satisfy the locality property that the points in the same cluster are close to each other. A number of clustering problems arise in machine learning where the optimal clusters do not follow such a locality property. Consider a variant of the $k$-means problem that may be regarded as a general version of such problems. Here, the optimal clusters $O_1, ..., O_k$ are an arbitrary partition of the dataset and the goal is to output $k$-centers $c_1, ..., c_k$ such that the objective function $\sum_{i=1}^{k} \sum_{x \in O_{i}} ||x - c_{i}||^2$ is minimized. It is not difficult to argue that any algorithm (without knowing the optimal clusters) that outputs a single set of $k$ centers, will not behave well as far as optimizing the above objective function is concerned. However, this does not rule out the existence of algorithms that output a list of such $k$ centers such that at least one of these $k$ centers behaves well. Given an error parameter $\varepsilon > 0$, let $\ell$ denote the size of the smallest list of $k$-centers such that at least one of the $k$-centers gives a $(1+\varepsilon)$ approximation w.r.t. the objective function above. In this paper, we show an upper bound on $\ell$ by giving a randomized algorithm that outputs a list of $2^{\tilde{O}(k/\varepsilon)}$ $k$-centers. We also give a closely matching lower bound of $2^{\tildeΩ(k/\sqrt{\varepsilon})}$. Moreover, our algorithm runs in time $O \left(n d \cdot 2^{\tilde{O}(k/\varepsilon)} \right)$. This is a significant improvement over the previous result of Ding and Xu who gave an algorithm with running time $O \left(n d \cdot (\log{n})^{k} \cdot 2^{poly(k/\varepsilon)} \right)$ and output a list of size $O \left((\log{n})^k \cdot 2^{poly(k/\varepsilon)} \right)$.
Faster Fundamental Graph Algorithms via Learned Predictions
我们考虑用机器学习预测加速经典图算法的问题。在这个模型中，算法配备了从过去或类似实例中学习的额外建议。鉴于额外的信息，我们的目标是改善传统的最坏情况下的运行时保证。我们的贡献如下：（i）我们给出了一个快速的最小权值二部匹配算法，改进了Dinitz，Im，Lavastida，Moseley和Vassilvitskii最近的结果（NeurIPS，2021）;（ii）将学习对偶方法推广到单源最短路径问题（具有负边缘长度），在给出足够精确的预测的情况下实现几乎线性的运行时间，这改进了由于Goldberg（SIAM J. Comput.，（1995年）;（iii）我们为基于学习的图算法提供了一个通用的基于约简的框架，基于对二分匹配和最短路径问题的约简，得到了度约束子图和最小代价$0$ - $1$流的新算法。最后，我们给出了一组一般的可学习性定理，表明我们的算法所需的预测可以有效地学习PAC的方式。
