Large-Scale Distributed Algorithms for Facility Location with Outliers

This paper presents fast, distributed, O(1)-approximation algorithms for metric facility location problems with outliers in the Congested Clique model, Massively Parallel Computation (MPC) model, and in the k-machine model. The paper considers Robust Facility Location and Facility Location with Penalties, two versions of the facility location problem with outliers proposed by Charikar et al. (SODA 2001). The paper also considers two alternatives for specifying the input: the input metric can be provided explicitly (as an n x n matrix distributed among the machines) or implicitly as the shortest path metric of a given edge-weighted graph. The results in the paper are: - Implicit metric: For both problems, O(1)-approximation algorithms running in O(poly(log n)) rounds in the Congested Clique and the MPC model and O(1)-approximation algorithms running in O~(n/k) rounds in the k-machine model. - Explicit metric: For both problems, O(1)-approximation algorithms running in O(log log log n) rounds in the Congested Clique and the MPC model and O(1)-approximation algorithms running in O~(n/k) rounds in the k-machine model. Our main contribution is to show the existence of Mettu-Plaxton-style O(1)-approximation algorithms for both Facility Location with outlier problems. As shown in our previous work (Berns et al., ICALP 2012, Bandyapadhyay et al., ICDCN 2018) Mettu-Plaxton style algorithms are more easily amenable to being implemented efficiently in distributed and large-scale models of computation

FPT Approximation and Subexponential Algorithms for Covering Few or Many Edges

We study the ?-Fixed Cardinality Graph Partitioning (?-FCGP) problem, the generic local graph partitioning problem introduced by Bonnet et al. [Algorithmica 2015]. In this problem, we are given a graph G, two numbers k,p and 0 ? ? ? 1, the question is whether there is a set S ? V of size k with a specified coverage function cov_?(S) at least p (or at most p for the minimization version). The coverage function cov_?(?) counts edges with exactly one endpoint in S with weight ? and edges with both endpoints in S with weight 1 - ?. ?-FCGP generalizes a number of fundamental graph problems such as Densest k-Subgraph, Max k-Vertex Cover, and Max (k,n-k)-Cut. A natural question in the study of ?-FCGP is whether the algorithmic results known for its special cases, like Max k-Vertex Cover, could be extended to more general settings. One of the simple but powerful methods for obtaining parameterized approximation [Manurangsi, SOSA 2019] and subexponential algorithms [Fomin et al. IPL 2011] for Max k-Vertex Cover is based on the greedy vertex degree orderings. The main insight of our work is that the idea of greed vertex degree ordering could be used to design fixed-parameter approximation schemes (FPT-AS) for ? > 0 and the subexponential-time algorithms for the problem on apex-minor free graphs for maximization with ? > 1/3 and minimization with ? < 1/3

Prediction of Air-Side Particulate Fouling of HVAC&R Heat Exchangers

Air-to-refrigerant heat exchangers used in heating, ventilation, air-conditioning, and refrigeration systems routinely experience air-side fouling due to the presence of particulates in outdoor and indoor environments. The influence on the performance of the heat exchanger, in terms of heat transfer efficiency and pressure drop imposed, depends on the extent of air-side fouling. Fouling of a heat exchanger is determined by a variety of parameters such as the dimensions of the heat exchanger, physical properties of the airborne particulates, and airflow conditions over the heat exchange surfaces. A comprehensive model is developed to deterministically calculate the extent of fouling of a heat exchanger as a function of these parameters by accounting for each of the possible deposition mechanisms. The study enhances modeling approaches previously employed in the literature by accounting for time-dependent accumulation of particles as well as the effects of the streamwise distribution of accumulated dust on subsequent fouling; the calculations for the deposition due to several of the mechanisms are also refined to improve prediction accuracy. Particulate matter deposits already present on the surface are found to accelerate the process of fouling by decreasing available area for airflow; an existing deposit layer effectively decreases the distance that a particle must travel to collide with a surface and increases the surface area available for deposition. The modified model predictions are compared against extant experimental deposition fraction data; an improved agreement is observed compared to previous models in the literature

Kernelization for Spreading Points

We consider the following problem about dispersing points. Given a set of points in the plane, the task is to identify whether by moving a small number of points by small distance, we can obtain an arrangement of points such that no pair of points is ``close" to each other. More precisely, for a family of $n$ points, an integer $k$, and a real number $d > 0$, we ask whether at most $k$ points could be relocated, each point at distance at most $d$ from its original location, such that the distance between each pair of points is at least a fixed constant, say $1$. A number of approximation algorithms for variants of this problem, under different names like distant representatives, disk dispersing, or point spreading, are known in the literature. However, to the best of our knowledge, the parameterized complexity of this problem remains widely unexplored. We make the first step in this direction by providing a kernelization algorithm that, in polynomial time, produces an equivalent instance with $O(d^2k^3)$ points. As a byproduct of this result, we also design a non-trivial fixed-parameter tractable (FPT) algorithm for the problem, parameterized by $k$ and $d$. Finally, we complement the result about polynomial kernelization by showing a lower bound that rules out the existence of a kernel whose size is polynomial in $k$ alone, unless $\mathsf{NP} \subseteq \mathsf{coNP}/\text{poly}$

Exact Exponential Algorithms for Clustering Problems

In this paper we initiate a systematic study of exact algorithms for some of the well known clustering problems, namely k-MEDIAN and k-MEANS. In k-MEDIAN, the input consists of a set X of n points belonging to a metric space, and the task is to select a subset C ? X of k points as centers, such that the sum of the distances of every point to its nearest center is minimized. In k-MEANS, the objective is to minimize the sum of squares of the distances instead. It is easy to design an algorithm running in time max_{k ? n} {n choose k} n^?(1) = ?^*(2?) (here, ?^*(?) notation hides polynomial factors in n). In this paper we design first non-trivial exact algorithms for these problems. In particular, we obtain an ?^*((1.89)?) time exact algorithm for k-MEDIAN that works for any value of k. Our algorithm is quite general in that it does not use any properties of the underlying (metric) space - it does not even require the distances to satisfy the triangle inequality. In particular, the same algorithm also works for k-Means. We complement this result by showing that the running time of our algorithm is asymptotically optimal, up to the base of the exponent. That is, unless the Exponential Time Hypothesis fails, there is no algorithm for these problems running in time 2^o(n)?n^?(1). Finally, we consider the "facility location" or "supplier" versions of these clustering problems, where, in addition to the set X we are additionally given a set of m candidate centers (or facilities) F, and objective is to find a subset of k centers from F. The goal is still to minimize the k-Median/k-Means/k-Center objective. For these versions we give a ?(2? (mn)^?(1)) time algorithms using subset convolution. We complement this result by showing that, under the Set Cover Conjecture, the "supplier" versions of these problems do not admit an exact algorithm running in time 2^{(1-?) n} (mn)^?(1)