Given two sets $S$ and $T$ of points in the plane, of total size $n$, a {many-to-many} matching between $S$ and $T$ is a set of pairs $(p,q)$ such that $pin S$, $qin T$ and for each $rin Scup T$, $r$ appears in at least one such pair. The {cost of a
pair} $(p,q)$ is the (Euclidean) distance between $p$ and $q$. In the {minimum-cost many-to-many matching} problem, the goal is to compute a many-to-many matching such that the sum of the costs of the pairs is minimized. This problem is a restricted version of minimum-weight edge cover in a bipartite graph, and hence can be solved in $O(n^3)$ time. In a more restricted setting where all the points are on a line, the problem can be solved in $O(nlog n)$ time [Colannino, Damian, Hurtado, Langerman, Meijer, Ramaswami, Souvaine, Toussaint; Graphs Comb., 2007]. However, no progress has been made in the general planar case in improving the cubic time bound. In this paper, we obtain an $O(n^2cdot poly(log n))$ time exact algorithm and an $O( n^{3/2}cdot poly(log n))$ time $(1+epsilon)$-approximation in the planar case. Our results affirmatively address an open problem posed in [Colannino et al., Graphs Comb., 2007].
In this paper, we consider the Minimum-Load $k$-Clustering/Facility Location (MLkC) problem where we are given a set $P$ of $n$ points in a metric space that we have to cluster and an integer $k$ that denotes the number of clusters. Additionally, we
are given a set $F$ of cluster centers in the same metric space. The goal is to select a set $Csubseteq F$ of $k$ centers and assign each point in $P$ to a center in $C$, such that the maximum load over all centers is minimized. Here the load of a center is the sum of the distances between it and the points assigned to it. Although clustering/facility location problems have a rich literature, the minimum-load objective is not studied substantially, and hence MLkC has remained a poorly understood problem. More interestingly, the problem is notoriously hard even in some special cases including the one in line metrics as shown by Ahmadian et al. [ACM Trans. Algo. 2018]. They also show APX-hardness of the problem in the plane. On the other hand, the best-known approximation factor for MLkC is $O(k)$, even in the plane. In this work, we study a fair version of MLkC inspired by the work of Chierichetti et al. [NeurIPS, 2017], which generalizes MLkC. Here the input points are colored by one of the $ell$ colors denoting the group they belong to. MLkC is the special case with $ell=1$. Considering this problem, we are able to obtain a $3$-approximation in $f(k,ell)cdot n^{O(1)}$ time. Also, our scheme leads to an improved $(1 + epsilon)$-approximation in case of Euclidean norm, and in this case, the running time depends only polynomially on the dimension $d$. Our results imply the same approximations for MLkC with running time $f(k)cdot n^{O(1)}$, achieving the first constant approximations for this problem in general and Euclidean metric spaces.
In this work, we study the $k$-median clustering problem with an additional equal-size constraint on the clusters, from the perspective of parameterized preprocessing. Our main result is the first lossy ($2$-approximate) polynomial kernel for this pr
oblem, parameterized by the cost of clustering. We complement this result by establishing lower bounds for the problem that eliminate the existences of an (exact) kernel of polynomial size and a PTAS.
We develop new algorithmic methods with provable guarantees for feature selection in regard to categorical data clustering. While feature selection is one of the most common approaches to reduce dimensionality in practice, most of the known feature s
election methods are heuristics. We study the following mathematical model. We assume that there are some inadvertent (or undesirable) features of the input data that unnecessarily increase the cost of clustering. Consequently, we want to select a subset of the original features from the data such that there is a small-cost clustering on the selected features. More precisely, for given integers $ell$ (the number of irrelevant features) and $k$ (the number of clusters), budget $B$, and a set of $n$ categorical data points (represented by $m$-dimensional vectors whose elements belong to a finite set of values $Sigma$), we want to select $m-ell$ relevant features such that the cost of any optimal $k$-clustering on these features does not exceed $B$. Here the cost of a cluster is the sum of Hamming distances ($ell_0$-distances) between the selected features of the elements of the cluster and its center. The clustering cost is the total sum of the costs of the clusters. We use the framework of parameterized complexity to identify how the complexity of the problem depends on parameters $k$, $B$, and $|Sigma|$. Our main result is an algorithm that solves the Feature Selection problem in time $f(k,B,|Sigma|)cdot m^{g(k,|Sigma|)}cdot n^2$ for some functions $f$ and $g$. In other words, the problem is fixed-parameter tractable parameterized by $B$ when $|Sigma|$ and $k$ are constants. Our algorithm is based on a solution to a more general problem, Constrained Clustering with Outliers. We also complement our algorithmic findings with complexity lower bounds.
In this paper, we study the lower- and upper-bounded covering (LUC) problem, where we are given a set $P$ of $n$ points, a collection $mathcal{B}$ of balls, and parameters $L$ and $U$. The goal is to find a minimum-sized subset $mathcal{B}subseteq ma
thcal{B}$ and an assignment of the points in $P$ to $mathcal{B}$, such that each point $pin P$ is assigned to a ball that contains $p$ and for each ball $B_iin mathcal{B}$, at least $L$ and at most $U$ points are assigned to $B_i$. We obtain an LP rounding based constant approximation for LUC by violating the lower and upper bound constraints by small constant factors and expanding the balls by again a small constant factor. Similar results were known before for covering problems with only the upper bound constraint. We also show that with only the lower bound constraint, the above result can be obtained without any lower bound violation. Covering problems have close connections with facility location problems. We note that the known constant-approximation for the corresponding lower- and upper-bounded facility location problem, violates the lower and upper bound constraints by a constant factor.
In the Metric Capacitated Covering (MCC) problem, given a set of balls $mathcal{B}$ in a metric space $P$ with metric $d$ and a capacity parameter $U$, the goal is to find a minimum sized subset $mathcal{B}subseteq mathcal{B}$ and an assignment of th
e points in $P$ to the balls in $mathcal{B}$ such that each point is assigned to a ball that contains it and each ball is assigned with at most $U$ points. MCC achieves an $O(log |P|)$-approximation using a greedy algorithm. On the other hand, it is hard to approximate within a factor of $o(log |P|)$ even with $beta < 3$ factor expansion of the balls. Bandyapadhyay~{et al.} [SoCG 2018, DCG 2019] showed that one can obtain an $O(1)$-approximation for the problem with $6.47$ factor expansion of the balls. An open question left by their work is to reduce the gap between the lower bound $3$ and the upper bound $6.47$. In this current work, we show that it is possible to obtain an $O(1)$-approximation with only $4.24$ factor expansion of the balls. We also show a similar upper bound of $5$ for a more generalized version of MCC for which the best previously known bound was $9$.
The Non-Uniform $k$-center (NUkC) problem has recently been formulated by Chakrabarty, Goyal and Krishnaswamy [ICALP, 2016] as a generalization of the classical $k$-center clustering problem. In NUkC, given a set of $n$ points $P$ in a metric space a
nd non-negative numbers $r_1, r_2, ldots , r_k$, the goal is to find the minimum dilation $alpha$ and to choose $k$ balls centered at the points of $P$ with radius $alphacdot r_i$ for $1le ile k$, such that all points of $P$ are contained in the union of the chosen balls. They showed that the problem is NP-hard to approximate within any factor even in tree metrics. On the other hand, they designed a bi-criteria constant approximation algorithm that uses a constant times $k$ balls. Surprisingly, no true approximation is known even in the special case when the $r_i$s belong to a fixed set of size 3. In this paper, we study the NUkC problem under perturbation resilience, which was introduced by Bilu and Linial [Combinatorics, Probability and Computing, 2012]. We show that the problem under 2-perturbation resilience is polynomial time solvable when the $r_i$s belong to a constant sized set. However, we show that perturbation resilience does not help in the general case. In particular, our findings imply that even with perturbation resilience one cannot hope to find any good approximation for the problem.
We study four classical graph problems -- Hamiltonian path, Traveling salesman, Minimum spanning tree, and Minimum perfect matching on geometric graphs induced by bichromatic (red and blue) points. These problems have been widely studied for points i
n the Euclidean plane, and many of them are NP-hard. In this work, we consider these problems in two restricted settings: (i) collinear points and (ii) equidistant points on a circle. We show that almost all of these problems can be solved in linear time in these constrained, yet non-trivial settings.
In this paper, we consider the colorful $k$-center problem, which is a generalization of the well-known $k$-center problem. Here, we are given red and blue points in a metric space, and a coverage requirement for each color. The goal is to find the s
mallest radius $rho$, such that with $k$ balls of radius $rho$, the desired number of points of each color can be covered. We obtain a constant approximation for this problem in the Euclidean plane. We obtain this result by combining a pseudo-approximation algorithm that works in any metric space, and an approximation algorithm that works for a special class of instances in the plane. The latter algorithm uses a novel connection to a certain matching problem in graphs.
In this article, we consider a collection of geometric problems involving points colored by two colors (red and blue), referred to as bichromatic problems. The motivation behind studying these problems is two fold; (i) these problems appear naturally
and frequently in the fields like Machine learning, Data mining, and so on, and (ii) we are interested in extending the algorithms and techniques for single point set (monochromatic) problems to bichromatic case. For all the problems considered in this paper, we design low polynomial time exact algorithms. These algorithms are based on novel techniques which might be of independent interest.
