5 research outputs found

### 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]