Minimum d-dimensional arrangement with fixed points

In the Minimum $d$-Dimensional Arrangement Problem (d-dimAP) we are given a graph with edge weights, and the goal is to find a 1-1 map of the vertices into $\mathbb{Z}^d$ (for some fixed dimension $d\geq 1$) minimizing the total weighted stretch of the edges. This problem arises in VLSI placement and chip design. Motivated by these applications, we consider a generalization of d-dimAP, where the positions of some of the vertices (pins) is fixed and specified as part of the input. We are asked to extend this partial map to a map of all the vertices, again minimizing the weighted stretch of edges. This generalization, which we refer to as d-dimAP+, arises naturally in these application domains (since it can capture blocked-off parts of the board, or the requirement of power-carrying pins to be in certain locations, etc.). Perhaps surprisingly, very little is known about this problem from an approximation viewpoint. For dimension $d=2$, we obtain an $O(k^{1/2} \cdot \log n)$-approximation algorithm, based on a strengthening of the spreading-metric LP for 2-dimAP. The integrality gap for this LP is shown to be $\Omega(k^{1/4})$. We also show that it is NP-hard to approximate 2-dimAP+ within a factor better than \Omega(k^{1/4-\eps}). We also consider a (conceptually harder, but practically even more interesting) variant of 2-dimAP+, where the target space is the grid $\mathbb{Z}_{\sqrt{n}} \times \mathbb{Z}_{\sqrt{n}}$, instead of the entire integer lattice $\mathbb{Z}^2$. For this problem, we obtain a $O(k \cdot \log^2{n})$-approximation using the same LP relaxation. We complement this upper bound by showing an integrality gap of $\Omega(k^{1/2})$, and an \Omega(k^{1/2-\eps})-inapproximability result. Our results naturally extend to the case of arbitrary fixed target dimension $d\geq 1$

Approximation Algorithms for Continuous Clustering and Facility Location Problems

We consider the approximability of center-based clustering problems where the points to be clustered lie in a metric space, and no candidate centers are specified. We call such problems "continuous", to distinguish from "discrete" clustering where candidate centers are specified. For many objectives, one can reduce the continuous case to the discrete case, and use an $\alpha$-approximation algorithm for the discrete case to get a $\beta\alpha$-approximation for the continuous case, where $\beta$ depends on the objective: e.g. for $k$-median, $\beta = 2$, and for $k$-means, $\beta = 4$. Our motivating question is whether this gap of $\beta$ is inherent, or are there better algorithms for continuous clustering than simply reducing to the discrete case? In a recent SODA 2021 paper, Cohen-Addad, Karthik, and Lee prove a factor-$2$ and a factor-$4$ hardness, respectively, for continuous $k$-median and $k$-means, even when the number of centers $k$ is a constant. The discrete case for a constant $k$ is exactly solvable in polytime, so the $\beta$ loss seems unavoidable in some regimes. In this paper, we approach continuous clustering via the round-or-cut framework. For four continuous clustering problems, we outperform the reduction to the discrete case. Notably, for the problem $\lambda$-UFL, where $\beta = 2$ and the discrete case has a hardness of $1.27$, we obtain an approximation ratio of $2.32 < 2 \times 1.27$ for the continuous case. Also, for continuous $k$-means, where the best known approximation ratio for the discrete case is $9$, we obtain an approximation ratio of $32 < 4 \times 9$. The key challenge is that most algorithms for discrete clustering, including the state of the art, depend on linear programs that become infinite-sized in the continuous case. To overcome this, we design new linear programs for the continuous case which are amenable to the round-or-cut framework.Comment: 24 pages, 0 figures. Full version of ESA 2022 paper https://drops.dagstuhl.de/opus/volltexte/2022/16971 . This version adds a link to the conference version and fixes minor formatting issue

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions

Hardness of Approximation for Euclidean k-Median

The Euclidean k-median problem is defined in the following manner: given a set ? of n points in d-dimensional Euclidean space ?^d, and an integer k, find a set C ? ?^d of k points (called centers) such that the cost function ?(C,?) ? ?_{x ? ?} min_{c ? C} ?x-c?? is minimized. The Euclidean k-means problem is defined similarly by replacing the distance with squared Euclidean distance in the cost function. Various hardness of approximation results are known for the Euclidean k-means problem [Pranjal Awasthi et al., 2015; Euiwoong Lee et al., 2017; Vincent Cohen{-}Addad and {Karthik {C. S.}}, 2019]. However, no hardness of approximation result was known for the Euclidean k-median problem. In this work, assuming the unique games conjecture (UGC), we provide the hardness of approximation result for the Euclidean k-median problem in O(log k) dimensional space. This solves an open question posed explicitly in the work of Awasthi et al. [Pranjal Awasthi et al., 2015]. Furthermore, we study the hardness of approximation for the Euclidean k-means/k-median problems in the bi-criteria setting where an algorithm is allowed to choose more than k centers. That is, bi-criteria approximation algorithms are allowed to output ? k centers (for constant ? > 1) and the approximation ratio is computed with respect to the optimal k-means/k-median cost. We show the hardness of bi-criteria approximation result for the Euclidean k-median problem for any ? < 1.015, assuming UGC. We also show a similar hardness of bi-criteria approximation result for the Euclidean k-means problem with a stronger bound of ? < 1.28, again assuming UGC

Matroid and Knapsack Center Problems

In the classic $k$-center problem, we are given a metric graph, and the objective is to open $k$ nodes as centers such that the maximum distance from any vertex to its closest center is minimized. In this paper, we consider two important generalizations of $k$-center, the matroid center problem and the knapsack center problem. Both problems are motivated by recent content distribution network applications. Our contributions can be summarized as follows: 1. We consider the matroid center problem in which the centers are required to form an independent set of a given matroid. We show this problem is NP-hard even on a line. We present a 3-approximation algorithm for the problem on general metrics. We also consider the outlier version of the problem where a given number of vertices can be excluded as the outliers from the solution. We present a 7-approximation for the outlier version. 2. We consider the (multi-)knapsack center problem in which the centers are required to satisfy one (or more) knapsack constraint(s). It is known that the knapsack center problem with a single knapsack constraint admits a 3-approximation. However, when there are at least two knapsack constraints, we show this problem is not approximable at all. To complement the hardness result, we present a polynomial time algorithm that gives a 3-approximate solution such that one knapsack constraint is satisfied and the others may be violated by at most a factor of $1+\epsilon$. We also obtain a 3-approximation for the outlier version that may violate the knapsack constraint by $1+\epsilon$.Comment: A preliminary version of this paper is accepted to IPCO 201

Constant-Factor FPT Approximation for Capacitated k-Median

Capacitated k-median is one of the few outstanding optimization problems for which the existence of a polynomial time constant factor approximation algorithm remains an open problem. In a series of recent papers algorithms producing solutions violating either the number of facilities or the capacity by a multiplicative factor were obtained. However, to produce solutions without violations appears to be hard and potentially requires different algorithmic techniques. Notably, if parameterized by the number of facilities k, the problem is also W[2] hard, making the existence of an exact FPT algorithm unlikely. In this work we provide an FPT-time constant factor approximation algorithm preserving both cardinality and capacity of the facilities. The algorithm runs in time 2^O(k log k) n^O(1) and achieves an approximation ratio of 7+epsilon

Better Guarantees for k-Means and Euclidean k-Median by Primal-Dual Algorithms

bibsource: dblp computer science bibliography, http://dblp.org biburl: http://dblp.org/rec/bib/conf/focs/AhmadianNSW17 timestamp: Thu, 16 Nov 2017 15:01:42 +0100 bdsk-url-1: https://doi.org/10.1109/FOCS.2017.15 bdsk-url-2: http://dx.doi.org/10.1109/FOCS.2017.15bibsource: dblp computer science bibliography, http://dblp.org biburl: http://dblp.org/rec/bib/conf/focs/AhmadianNSW17 timestamp: Thu, 16 Nov 2017 15:01:42 +0100 bdsk-url-1: https://doi.org/10.1109/FOCS.2017.15 bdsk-url-2: http://dx.doi.org/10.1109/FOCS.2017.1