We generalise the results of Bhattacharya et al. (Journal of Computing Systems, 62(1):93-115, 2018) for the list-$k$-means problem defined as -- for a (unknown) partition $X_1, ..., X_k$ of the dataset $X \subseteq \mathbb{R}^d$, find a list of $k$-center sets (each element in the list is a set of $k$ centers) such that at least one of $k$-center sets $\{c_1, ..., c_k\}$ in the list gives an $(1+\varepsilon)$-approximation with respect to the cost function $\min_{\textrm{permutation } π} \left[ \sum_{i=1}^{k} \sum_{x \in X_i} ||x - c_{π(i)}||^2 \right]$. The list-$k$-means problem is important for the constrained $k$-means problem since algorithms for the former can be converted to PTAS for various versions of the latter. Following are the consequences of our generalisations:
- Streaming algorithm: Our $D^2$-sampling based algorithm running in a single iteration allows us to design a 2-pass, logspace streaming algorithm for the list-$k$-means problem. This can be converted to a 4-pass, logspace streaming PTAS for various constrained versions of the $k$-means problem.
- Faster PTAS under stability: Our generalisation is also useful in $k$-means clustering scenarios where finding good centers becomes easy once good centers for a few "bad" clusters have been chosen. One such scenario is clustering under stability where the number of such bad clusters is a constant. Using the above idea, we significantly improve the running time of the known algorithm from $O(dn^3) (k \log{n})^{poly(\frac{1}β, \frac{1}{\varepsilon})}$ to $O \left(dn^3 k^{\tilde{O}_{β\varepsilon}(\frac{1}{β\varepsilon})} \right)$.
Space Lower Bounds for Approximating Maximum Matching in the Edge Arrival Model
在线和流媒体环境中的二分匹配问题最近受到了很多关注。经典的顶点到达设置，著名的Karp，Vazirani和Vazirani（KVV）算法实现了$1-1/e$近似，这是相当好理解的：$1-1/e$近似在在线和半流设置中都是最优的，其中算法被约束为使用$n\cdot \log^{O(1)} n$空间。更具有挑战性的边缘到达模型最近在在线算法文献中取得了重大进展。对于严格在线模型（无抢占），已排除优于平凡因子$1/2$的近似值[Gamlath et al'FOCS'19]。对于限制较少的在线抢先模型，已经排除了优于$\frac1{1+\ln 2}$ -近似[Epstein et al'STACS'12]甚至优于$(2-\sqrt{2})$ -近似[Huang et al'SODA'19]的情况。边缘到达模型中的在线抢先匹配的最新硬度结果是基于使用边缘到达将KVV硬实例的多个副本串在一起的想法。在本文中，我们将展示如何实现这样的建设使用的想法在Ruzsa-Szemerédi图的文献中。其结果是，我们表明，任何单程流算法，近似的最大匹配的二分图与$n$顶点的一个因素比$\frac1{1+\ln 2}\approx 0.59$需要$n^{1+Ω(1/\log\log n)}\gg n \log^{O(1)} n$空间。这给出了经典的单侧顶点到达设置和半流模型中的边到达设置之间的第一分离。
Improved Streaming Algorithm for Fair $k$-Center Clustering
School of Mathematics and Statistics, Fuzhou University（福州大学数学与统计学院）；School of Integrated Circuits, Guangdong University of Technology（广东工业大学集成电路学院）；School of Accounting, Information Systems and Supply Chain, RMIT University（皇家墨尔本理工大学会计、信息系统和供应链学院）
