A scalable solution for the extended multi-channel facility location problem

We study the extended version of the non-uniform, capacitated facility location problem with multiple fulfilment channels between the facilities and clients, each with their own channel capacities and service cost. Though the problem has been extensively studied in the literature, all the prior works assume a single channel of fulfilment, and the existing methods based on linear programming, primal-dual relationships, local search heuristics etc. do not scale for a large supply chain system involving millions of decision variables. Using the concepts of sub-modularity and optimal transport theory, we present a scalable algorithm for determining the set of facilities to be opened under a cardinality constraint. By introducing various schemes such as: (i) iterative facility selection using incremental gain, (ii) approximation of the linear program using novel multi-stage Sinkhorn iterations, (iii) creation of facilities one for each fulfilment channel etc., we develop a fast but a tight approximate solution, requiring $\mathcal{O}\left(\frac{3+k}{m}ln\left(\frac{1}{\epsilon}\right)\right)$ instances of optimal transport problems to select k facilities from m options, each solvable in linear time. Our algorithm is implicitly endowed with all the theoretical guarantees enjoyed by submodular maximisation problems and the Sinkhorn distances. When compared against the state-of-the-art commercial MILP solvers, we obtain a 100-fold speedup in computation, while the difference in objective values lies within a narrow range of 3%

Constant Factor Approximation for Capacitated k-Center with Outliers

The $k$-center problem is a classic facility location problem, where given an edge-weighted graph $G = (V,E)$ one is to find a subset of $k$ vertices $S$, such that each vertex in $V$ is "close" to some vertex in $S$. The approximation status of this basic problem is well understood, as a simple 2-approximation algorithm is known to be tight. Consequently different extensions were studied. In the capacitated version of the problem each vertex is assigned a capacity, which is a strict upper bound on the number of clients a facility can serve, when located at this vertex. A constant factor approximation for the capacitated $k$-center was obtained last year by Cygan, Hajiaghayi and Khuller [FOCS'12], which was recently improved to a 9-approximation by An, Bhaskara and Svensson [arXiv'13]. In a different generalization of the problem some clients (denoted as outliers) may be disregarded. Here we are additionally given an integer $p$ and the goal is to serve exactly $p$ clients, which the algorithm is free to choose. In 2001 Charikar et al. [SODA'01] presented a 3-approximation for the $k$-center problem with outliers. In this paper we consider a common generalization of the two extensions previously studied separately, i.e. we work with the capacitated $k$-center with outliers. We present the first constant factor approximation algorithm with approximation ratio of 25 even for the case of non-uniform hard capacities.Comment: 15 pages, 3 figures, accepted to STACS 201

Centrality of Trees for Capacitated k-Center

There is a large discrepancy in our understanding of uncapacitated and capacitated versions of network location problems. This is perhaps best illustrated by the classical k-center problem: there is a simple tight 2-approximation algorithm for the uncapacitated version whereas the first constant factor approximation algorithm for the general version with capacities was only recently obtained by using an intricate rounding algorithm that achieves an approximation guarantee in the hundreds. Our paper aims to bridge this discrepancy. For the capacitated k-center problem, we give a simple algorithm with a clean analysis that allows us to prove an approximation guarantee of 9. It uses the standard LP relaxation and comes close to settling the integrality gap (after necessary preprocessing), which is narrowed down to either 7, 8 or 9. The algorithm proceeds by first reducing to special tree instances, and then solves such instances optimally. Our concept of tree instances is quite versatile, and applies to natural variants of the capacitated k-center problem for which we also obtain improved algorithms. Finally, we give evidence to show that more powerful preprocessing could lead to better algorithms, by giving an approximation algorithm that beats the integrality gap for instances where all non-zero capacities are uniform.Comment: 21 pages, 2 figure

Capacitated Center Problems with Two-Sided Bounds and Outliers

In recent years, the capacitated center problems have attracted a lot of research interest. Given a set of vertices $V$, we want to find a subset of vertices $S$, called centers, such that the maximum cluster radius is minimized. Moreover, each center in $S$ should satisfy some capacity constraint, which could be an upper or lower bound on the number of vertices it can serve. Capacitated $k$-center problems with one-sided bounds (upper or lower) have been well studied in previous work, and a constant factor approximation was obtained. We are the first to study the capacitated center problem with both capacity lower and upper bounds (with or without outliers). We assume each vertex has a uniform lower bound and a non-uniform upper bound. For the case of opening exactly $k$ centers, we note that a generalization of a recent LP approach can achieve constant factor approximation algorithms for our problems. Our main contribution is a simple combinatorial algorithm for the case where there is no cardinality constraint on the number of open centers. Our combinatorial algorithm is simpler and achieves better constant approximation factor compared to the LP approach

An Improved Approximation Algorithm for the Hard Uniform Capacitated k-median Problem

In the $k$-median problem, given a set of locations, the goal is to select a subset of at most $k$ centers so as to minimize the total cost of connecting each location to its nearest center. We study the uniform hard capacitated version of the $k$-median problem, in which each selected center can only serve a limited number of locations. Inspired by the algorithm of Charikar, Guha, Tardos and Shmoys, we give a $(6+10\alpha)$-approximation algorithm for this problem with increasing the capacities by a factor of $2+\frac{2}{\alpha}, \alpha\geq 4$, which improves the previous best $(32 l^2+28 l+7)$-approximation algorithm proposed by Byrka, Fleszar, Rybicki and Spoerhase violating the capacities by factor $2+\frac{3}{l-1}, l\in \{2,3,4,\dots\}$.Comment: 19 pages, 1 figur

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