FPT Approximations for Capacitated/Fair Clustering with Outliers

Clustering problems such as $k$-Median, and $k$-Means, are motivated from applications such as location planning, unsupervised learning among others. In such applications, it is important to find the clustering of points that is not ``skewed'' in terms of the number of points, i.e., no cluster should contain too many points. This is modeled by capacity constraints on the sizes of clusters. In an orthogonal direction, another important consideration in clustering is how to handle the presence of outliers in the data. Indeed, these clustering problems have been generalized in the literature to separately handle capacity constraints and outliers. To the best of our knowledge, there has been very little work on studying the approximability of clustering problems that can simultaneously handle both capacities and outliers. We initiate the study of the Capacitated $k$-Median with Outliers (C$k$MO) problem. Here, we want to cluster all except $m$ outlier points into at most $k$ clusters, such that (i) the clusters respect the capacity constraints, and (ii) the cost of clustering, defined as the sum of distances of each non-outlier point to its assigned cluster-center, is minimized. We design the first constant-factor approximation algorithms for C$k$MO. In particular, our algorithm returns a (3+\epsilon)-approximation for C$k$MO in general metric spaces, and a (1+\epsilon)-approximation in Euclidean spaces of constant dimension, that runs in time in time $f(k, m, \epsilon) \cdot |I_m|^{O(1)}$, where $|I_m|$ denotes the input size. We can also extend these results to a broader class of problems, including Capacitated k-Means/k-Facility Location with Outliers, and Size-Balanced Fair Clustering problems with Outliers. For each of these problems, we obtain an approximation ratio that matches the best known guarantee of the corresponding outlier-free problem.Comment: Abstract shortened to meet arxiv requirement

Near-optimal Algorithms for Stochastic Online Bin Packing

We study the online bin packing problem under two stochastic settings. In the bin packing problem, we are given n items with sizes in (0,1] and the goal is to pack them into the minimum number of unit-sized bins. First, we study bin packing under the i.i.d. model, where item sizes are sampled independently and identically from a distribution in (0,1]. Both the distribution and the total number of items are unknown. The items arrive one by one and their sizes are revealed upon their arrival and they must be packed immediately and irrevocably in bins of size 1. We provide a simple meta-algorithm that takes an offline $\alpha$-asymptotic approximation algorithm and provides a polynomial-time $(\alpha + \varepsilon)$-competitive algorithm for online bin packing under the i.i.d. model, where $\varepsilon$>0 is a small constant. Using the AFPTAS for offline bin packing, we thus provide a linear time $(1+\varepsilon)$-competitive algorithm for online bin packing under i.i.d. model, thus settling the problem. We then study the random-order model, where an adversary specifies the items, but the order of arrival of items is drawn uniformly at random from the set of all permutations of the items. Kenyon's seminal result [SODA'96] showed that the Best-Fit algorithm has a competitive ratio of at most 3/2 in the random-order model, and conjectured the ratio to be around 1.15. However, it has been a long-standing open problem to break the barrier of 3/2 even for special cases. Recently, Albers et al. [Algorithmica'21] showed an improvement to 5/4 competitive ratio in the special case when all the item sizes are greater than 1/3. For this special case, we settle the analysis by showing that Best-Fit has a competitive ratio of 1. We make further progress by breaking the barrier of 3/2 for the 3-Partition problem, a notoriously hard special case of bin packing, where all item sizes lie in (1/4,1/2]