The online unit clustering problem was proposed by Chan and Zarrabi-Zadeh (WAOA2007 and Theory of Computing Systems 45(3), 2009), which is defined as follows: "Points" are given online in the $d$-dimensional Euclidean space one by one. An algorithm creates a "cluster," which is a $d$-dimensional rectangle. The initial length of each edge of a cluster is 0. An algorithm can extend an edge until it reaches unit length independently of other dimensions. The task of an algorithm is to cover a new given point either by creating a new cluster and assigning it to the point, or by extending edges of an existing cluster created in past times. The goal is to minimize the total number of created clusters. Chan and Zarrabi-Zadeh proposed some method to obtain a competitive algorithm for the $d$-dimensional case using an algorithm for the one-dimensional case, and thus the one-dimensional case has been extensively studied including some variants of the unit clustering problem. In this paper, we show a lower bound of $13/8 = 1.625$ on the competitive ratio of any deterministic online algorithm for the one-dimensional unit clustering, improving the previous lower bound $8/5 (=1.6)$ presented by Epstein and van Stee (WAOA2007 and ACM Transactions on Algorithms 7(1), 2010). Note that Ehmsen and Larsen (SWAT2010 and Theoretical Computer Science, 500, 2013) showed the current best upper bound of $5/3$, and conjectured that the exact competitive ratio in the one-dimensional case may be $13/8$.
We study the online clustering problem where data items arrive in an online fashion. The algorithm maintains a clustering of data items into similarity classes. Upon arrival of v, the relation between v and previously arrived items is revealed, so that for each u we are told whether v is similar to u. The algorithm can create a new cluster for v and merge existing clusters. When the objective is to minimize disagreements between the clustering and the input, we prove that a natural greedy algorithm is O(n)-competitive, and this is optimal. When the objective is to maximize agreements between t…
In Online Sum-Radii Clustering, n demand points arrive online and must be irrevocably assigned to a cluster upon arrival. The cost of each cluster is the sum of a fixed opening cost and its radius, and the objective is to minimize the total cost of the clusters opened by the algorithm. We show that the deterministic competitive ratio of Online Sum-Radii Clustering for general metric spaces is Θ(\log n), where the upper bound follows from a primal-dual algorithm and holds for general metric spaces, and the lower bound is valid for ternary Hierarchically Well-Separated Trees (HSTs) and for the E…
Clique clustering is the problem of partitioning the vertices of a graph into disjoint clusters, where each cluster forms a clique in the graph, while optimizing some objective function. In online clustering, the input graph is given one vertex at a time, and any vertices that have previously been clustered together are not allowed to be separated. The goal is to maintain a clustering with an objective value close to the optimal solution. For the variant where we want to maximize the number of edges in the clusters, we propose an online strategy based on the doubling technique. It has an asymp…
This paper shows that one can be competitive with the k-means objective while operating online. In this model, the algorithm receives vectors v_1,...,v_n one by one in an arbitrary order. For each vector the algorithm outputs a cluster identifier before receiving the next one. Our online algorithm generates ~O(k) clusters whose k-means cost is ~O(W*). Here, W* is the optimal k-means cost using k clusters and ~O suppresses poly-logarithmic factors. We also show that, experimentally, it is not much worse than k-means++ while operating in a strictly more constrained computational model.
In the online balanced graph repartitioning problem, one has to maintain a clustering of $n$ nodes into $\ell$ clusters, each having $k = n / \ell$ nodes. During runtime, an online algorithm is given a stream of communication requests between pairs of nodes: an inter-cluster communication costs one unit, while the intra-cluster communication is free. An algorithm can change the clustering, paying unit cost for each moved node. This natural problem admits a simple $O(\ell^2 \cdot k^2)$-competitive algorithm COMP, whose performance is far apart from the best known lower bound of $Ω(\ell \cdot k)…
Offline k-means clustering was studied extensively, and algorithms with a constant approximation are available. However, online clustering is still uncharted. New factors come into play: the ordering of the dataset and whether the number of points, n, is known in advance or not. Their exact effects are unknown. In this paper we focus on the online setting where the decisions are irreversible: after a point arrives, the algorithm needs to decide whether to take the point as a center or not, and this decision is final. How many centers are needed and sufficient to achieve constant approximation …
IIT Gandhinagar（印度理工学院甘地讷格尔分校）；University of Warwick（华威大学）
We present algorithms that create coresets in an online setting for clustering problems according to a wide subset of Bregman divergences. Notably, our coresets have a small additive error, similar in magnitude to the lightweight coresets Bachem et. al. 2018, and take update time $O(d)$ for every incoming point where $d$ is dimension of the point. Our first algorithm gives online coresets of size $\tilde{O}(\mbox{poly}(k,d,ε,μ))$ for $k$-clusterings according to any $μ$-similar Bregman divergence. We further extend this algorithm to show existence of a non-parametric coresets, where the corese…
We consider online $k$-means clustering where each new point is assigned to the nearest cluster center, after which the algorithm may update its centers. The loss incurred is the sum of squared distances from new points to their assigned cluster centers. The goal over a data stream $X$ is to achieve loss that is a constant factor of $L(X, OPT_k)$, the best possible loss using $k$ fixed points in hindsight. We propose a data parameter, $Λ(X)$, such that for any algorithm maintaining $O(k\text{poly}(\log n))$ centers at time $n$, there exists a data stream $X$ for which a loss of $Ω(Λ(X))$ is in…
In the correlation clustering problem for complete signed graphs, the input is a complete signed graph with edges weighted as $+1$ (denote recommendation to put this pair in the same cluster) or $-1$ (recommending to put this pair of vertices in separate clusters) and the target is to cluster the set of vertices such that the number of disagreements with these recommendations is minimized. In this paper, we consider the problem of correlation clustering for dynamic complete signed graphs where (1) a vertex can be added or deleted, and (2) the sign of an edge can be flipped. In the proposed onl…
Kierstead and Trotter (Congressus Numerantium 33, 1981) proved that their algorithm is an optimal online algorithm for the online interval coloring problem. In this paper, for online unit interval coloring, we show that the number of colors used by the Kierstead-Trotter algorithm is at most $3 ω(G) - 3$, where $ω(G)$ is the size of the maximum clique in a given graph $G$, and it is the best possible.
Online Correlation Clustering: Simultaneously Optimizing All $\ell_p$-norms
Department of EECS at UC Berkeley（伯克利大学电子工程与计算机科学系）；RelationalAI；Tepper School of Business, Carnegie Mellon University（卡内基梅隆大学泰珀商学院）；Carnegie Mellon University（卡内基梅隆大学）；Department of Computer Science, Vassar College（瓦萨学院计算机科学系）
A variant of the online knapsack problem is considered in the settings of trusted and untrusted predictions. In Unit Profit Knapsack, the items have unit profit, and it is easy to find an optimal solution offline: Pack as many of the smallest items as possible into the knapsack. For Online Unit Profit Knapsack, the competitive ratio is unbounded. In contrast, previous work on online algorithms with untrusted predictions generally studied problems where an online algorithm with a constant competitive ratio is known. The prediction, possibly obtained from a machine learning source, that our algo…
We improve the best known competitive ratio (from 1/4 to 1/2), for the online multi-unit allocation problem, where the objective is to maximize the single-price revenue. Moreover, the competitive ratio of our algorithm tends to 1, as the bid-profile tends to ``smoothen''. This algorithm is used as a subroutine in designing truthful auctions for the same setting: the allocation has to be done online, while the payments can be decided at the end of the day. Earlier, a reduction from the auction design problem to the allocation problem was known only for the unit-demand case. We give a reduction …
Università degli Studi di Milano（米兰大学）；Università dell’Insubria（因苏布里亚大学）
Motivated by social balance theory, we develop a theory of link classification in signed networks using the correlation clustering index as measure of label regularity. We derive learning bounds in terms of correlation clustering within three fundamental transductive learning settings: online, batch and active. Our main algorithmic contribution is in the active setting, where we introduce a new family of efficient link classifiers based on covering the input graph with small circuits. These are the first active algorithms for link classification with mistake bounds that hold for arbitrary sign…
