We consider the {\em matroid median} problem \cite{KrishnaswamyKNSS11}, wherein we are given a set of facilities with opening costs and a matroid on the facility-set, and clients with demands and connection costs, and we seek to open an independent set of facilities and assign clients to open facilities so as to minimize the sum of the facility-opening and client-connection costs. We give a simple 8-approximation algorithm for this problem based on LP-rounding, which improves upon the 16-approximation in \cite{KrishnaswamyKNSS11}. Our techniques illustrate that much of the work involved in the rounding algorithm of in \cite{KrishnaswamyKNSS11} can be avoided by first converting the LP solution to a half-integral solution, which can then be rounded to an integer solution using a simple uncapacitated-facility-location (UFL) style clustering step. We illustrate the power of these ideas by deriving: (a) a 24-approximation algorithm for matroid median with penalties, which is a vast improvement over the 360-approximation obtained in \cite{KrishnaswamyKNSS11}; and (b) an 8-approximation for the {\em two-matroid median} problem, a generalization of matroid median that we introduce involving two matroids. We show that a variety of seemingly disparate facility-location problems considered in the literature---data placement problem, mobile facility location, $k$-median forest, metric uniform minimum-latency UFL---in fact reduce to the matroid median or two-matroid median problems, and thus obtain improved approximation guarantees for all these problems. Our techniques also yield an improvement for the knapsack median problem.
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 proble…
Consider the following problem: given a set system (U,I) and an edge-weighted graph G = (U, E) on the same universe U, find the set A in I such that the Steiner tree cost with terminals A is as large as possible: "which set in I is the most difficult to connect up?" This is an example of a max-min problem: find the set A in I such that the value of some minimization (covering) problem is as large as possible. In this paper, we show that for certain covering problems which admit good deterministic online algorithms, we can give good algorithms for max-min optimization when the set system I is g…
Submodular function maximization is a central problem in combinatorial optimization, generalizing many important problems including Max Cut in directed/undirected graphs and in hypergraphs, certain constraint satisfaction problems, maximum entropy sampling, and maximum facility location problems. Unlike submodular minimization, submodular maximization is NP-hard. For the problem of maximizing a non-monotone submodular function, Feige, Mirrokni, and Vondrák recently developed a $2\over 5$-approximation algorithm \cite{FMV07}, however, their algorithms do not handle side constraints.} In this pa…
We consider a problem of maximizing a monotone DR-submodular function under multiple order-consistent knapsack constraints on a distributive lattice. Since a distributive lattice is used to represent a dependency constraint, the problem can represent a dependency constrained version of a submodular maximization problem on a set. We propose a $1 - 1/e$ approximation algorithm for this problem. To achieve this result, we generalize the continuous greedy algorithm to distributive lattices: We choose a median complex as a continuous relaxation of a distributive lattice and define the multilinear e…
In this paper, we study the fault-tolerant matroid median and fault-tolerant knapsack median problems. These two problems generalize many fundamental clustering and facility location problems, such as uniform fault-tolerant $k$-median, uniform fault-tolerant facility location, matroid median, knapsack median, etc. We present a versatile iterative rounding framework and obtain a unifying constant-factor approximation algorithm.
This paper considers approximation algorithms for generalized $k$-median problems. This class of problems can be informally described as $k$-median with a constant number of extra constraints, and includes $k$-median with outliers, and knapsack median. Our first contribution is a pseudo-approximation algorithm for generalized $k$-median that outputs a $6.387$-approximate solution, with a constant number of fractional variables. The algorithm builds on the iterative rounding framework introduced by Krishnaswamy, Li, and Sandeep for $k$-median with outliers. The main technical innovation is allo…
In the matroid center problem, which generalizes the $k$-center problem, we need to pick a set of centers that is an independent set of a matroid with rank $r$. We study this problem in streaming, where elements of the ground set arrive in the stream. We first show that any randomized one-pass streaming algorithm that computes a better than $Δ$-approximation for partition-matroid center must use $Ω(r^2)$ bits of space, where $Δ$ is the aspect ratio of the metric and can be arbitrarily large. This shows a quadratic separation between matroid center and $k$-center, for which the Doubling algorit…
We consider the budgeted matroid independent set problem. The input is a ground set, where each element has a cost and a non-negative profit, along with a matroid over the elements and a budget. The goal is to select a subset of elements which maximizes the total profit subject to the matroid and budget constraints. Several well known special cases, where we have, e.g., a uniform matroid and a budget, or no matroid constraint (i.e., the classic knapsack problem), admit a fully polynomial-time approximation scheme (FPTAS). In contrast, already a slight generalization to the multi-budgeted matro…
We present a simple combinatorial $\frac{1 -e^{-2}}{2}$-approximation algorithm for maximizing a monotone submodular function subject to a knapsack and a matroid constraint. This classic problem is known to be hard to approximate within factor better than $1 - 1/e$. We show that the algorithm can be extended to yield a ratio of $\frac{1 - e^{-(k+1)}}{k+1}$ for the problem with a single knapsack and the intersection of $k$ matroid constraints, for any fixed $k > 1$. Our algorithms, which combine the greedy algorithm of [Khuller, Moss and Naor, 1999] and [Sviridenko, 2004] with local search, sho…
We study the $k$-median with discounts problem, wherein we are given clients with non-negative discounts and seek to open at most $k$ facilities. The goal is to minimize the sum of distances from each client to its nearest open facility which is discounted by its own discount value, with minimum contribution being zero. $k$-median with discounts unifies many classic clustering problems, e.g., $k$-center, $k$-median, $k$-facility $l$-centrum, etc. We obtain a bi-criteria constant-factor approximation using an iterative LP rounding algorithm. Our result improves the previously best approximation…
We study the problem of maximizing constrained non-monotone submodular functions and provide approximation algorithms that improve existing algorithms in terms of either the approximation factor or simplicity. Our algorithms combine existing local search and greedy based algorithms. Different constraints that we study are exact cardinality and multiple knapsack constraints. For the multiple-knapsack constraints we achieve a $(0.25-2ε)$-factor algorithm. We also show, as our main contribution, how to use the continuous greedy process for non-monotone functions and, as a result, obtain a $0.13$-…
In this paper, we study some multiagent variants of the knapsack problem. Fluschnik et al. [AAAI 2019] considered the model in which every agent assigns some utility to every item. They studied three preference aggregation rules for finding a subset (knapsack) of items: individually best, diverse, and Nash-welfare-based. Informally, diversity is achieved by satisfying as many voters as possible. Motivated by the application of aggregation operators in multiwinner elections, we extend the study from diverse aggregation rule to Median and Best scoring functions. We study the computational and pa…
Like most multiobjective combinatorial optimization problems, biobjective optimization problems on matroids are in general intractable and their corresponding decision problems are in general NP-hard. In this paper, we consider biobjective optimization problems on matroids where one of the objective functions is restricted to binary cost coefficients. We show that in this case the problem has a connected efficient set with respect to a natural definition of a neighborhood structure and hence, can be solved efficiently using a neighborhood search approach. This is, to the best of our knowledge,…
We pursue a study of the Generalized Demand Matching problem, a common generalization of the $b$-Matching and Knapsack problems. Here, we are given a graph with vertex capacities, edge profits, and asymmetric demands on the edges. The goal is to find a maximum-profit subset of edges so the demands of chosen edges do not violate vertex capacities. This problem is APX-hard and constant-factor approximations are known. Our results fall into two categories. First, using iterated relaxation and various filtering strategies, we show with an efficient rounding algorithm if an additional matroid struc…
We develop approximation algorithms for set-selection problems with deterministic constraints, but random objective values, i.e., stochastic probing problems. When the goal is to maximize the objective, approximation algorithms for probing problems are well-studied. On the other hand, few techniques are known for minimizing the objective, especially in the adaptive setting, where information about the random objective is revealed during the set-selection process and allowed to influence it. For minimization problems in particular, incorporating adaptivity can have a considerable effect on perf…
We study a general stochastic probing problem defined on a universe V, where each element e in V is "active" independently with probability p_e. Elements have weights {w_e} and the goal is to maximize the weight of a chosen subset S of active elements. However, we are given only the p_e values-- to determine whether or not an element e is active, our algorithm must probe e. If element e is probed and happens to be active, then e must irrevocably be added to the chosen set S; if e is not active then it is not included in S. Moreover, the following conditions must hold in every random instantiat…
In this paper, we present a new iterative rounding framework for many clustering problems. Using this, we obtain an $(α_1 + ε\leq 7.081 + ε)$-approximation algorithm for $k$-median with outliers, greatly improving upon the large implicit constant approximation ratio of Chen [Chen, SODA 2018]. For $k$-means with outliers, we give an $(α_2+ε\leq 53.002 + ε)$-approximation, which is the first $O(1)$-approximation for this problem. The iterative algorithm framework is very versatile; we show how it can be used to give $α_1$- and $(α_1 + ε)$-approximation algorithms for matroid and knapsack median …
In this paper, we study the following robust optimization problem. Given an independence system and candidate objective functions, we choose an independent set, and then an adversary chooses one objective function, knowing our choice. Our goal is to find a randomized strategy (i.e., a probability distribution over the independent sets) that maximizes the expected objective value. To solve the problem, we propose two types of schemes for designing approximation algorithms. One scheme is for the case when objective functions are linear. It first finds an approximately optimal aggregated strategy…
In this paper, we study two natural generalizations of ordered $k$-median, named robust ordered $k$-median and fault-tolerant ordered $k$-median. In ordered $k$-median, given a finite metric space $(X,d)$, we seek to open $k$ facilities $S\subseteq X$ which induce a service cost vector $\vec{c}=\{d(j,S):j\in X\}$, and minimize the ordered objective $w^\top\vec{c}^\downarrow$. Here $d(j,S)=\min_{i\in S}d(j,i)$ is the minimum distance between $j$ and facilities in $S$, $w\in\mathbb{R}^{|X|}$ is a given non-increasing non-negative vector, and $\vec{c}^\downarrow$ is the non-increasingly sorted ve…
Fair clustering enjoyed a surge of interest recently. One appealing way of integrating fairness aspects into classical clustering problems is by introducing multiple covering constraints. This is a natural generalization of the robust (or outlier) setting, which has been studied extensively and is amenable to a variety of classic algorithmic techniques. In contrast, for the case of multiple covering constraints (the so-called colorful setting), specialized techniques have only been developed recently for $k$-Center clustering variants, which is also the focus of this paper. While prior techniq…
Constrained submodular maximization problems have long been studied, with near-optimal results known under a variety of constraints when the submodular function is monotone. The case of non-monotone submodular maximization is less understood: the first approximation algorithms even for the unconstrainted setting were given by Feige et al. (FOCS '07). More recently, Lee et al. (STOC '09, APPROX '09) show how to approximately maximize non-monotone submodular functions when the constraints are given by the intersection of p matroid constraints; their algorithm is based on local-search procedures …
We introduce a novel problem for diversity-aware clustering. We assume that the potential cluster centers belong to a set of groups defined by protected attributes, such as ethnicity, gender, etc. We then ask to find a minimum-cost clustering of the data into $k$ clusters so that a specified minimum number of cluster centers are chosen from each group. We thus require that all groups are represented in the clustering solution as cluster centers, according to specified requirements. More precisely, we are given a set of clients $C$, a set of facilities $\pazocal{F}$, a collection $\mathcal{F}=\…
A $(2+\varepsilon)$-Approximation Algorithm for Metric $k$-Median
Google Research, France（谷歌法国研究院）；IDSIA, USI-SUPSI, Switzerland（瑞士IDSIA、USI-SUPSI）；University of Michigan, Ann Arbor, MI, USA（美国密歇根大学安娜堡分校）；Aarhus University, Denmark（丹麦奥胡斯大学）；EPFL, Switzerland（瑞士洛桑联邦理工学院）
Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）；Technical University of Dortmund（多特蒙德工业大学）
We consider the $k$-clustering problem with $\ell_p$-norm cost, which includes $k$-median, $k$-means and $k$-center, under an individual notion of fairness proposed by Jung et al. [2020]: given a set of points $P$ of size $n$, a set of $k$ centers induces a fair clustering if every point in $P$ has a center among its $n/k$ closest neighbors. Mahabadi and Vakilian [2020] presented a $(p^{O(p)},7)$-bicriteria approximation for fair clustering with $\ell_p$-norm cost: every point finds a center within distance at most $7$ times its distance to its $(n/k)$-th closest neighbor and the $\ell_p$-norm…
Constrained submodular function maximization has been used in subset selection problems such as selection of most informative sensor locations. While these models have been quite popular, the solutions Constrained submodular function maximization has been used in subset selection problems such as selection of most informative sensor locations. While these models have been quite popular, the solutions obtained via this approach are unstable to perturbations in data defining the submodular functions. Robust submodular maximization has been proposed as a richer model that aims to overcome this di…
Clustering with outliers is one of the most fundamental problems in Computer Science. Given a set $X$ of $n$ points and two integers $k$ and $m$, the clustering with outliers aims to exclude $m$ points from $X$ and partition the remaining points into $k$ clusters that minimizes a certain cost function. In this paper, we give a general approach for solving clustering with outliers, which results in a fixed-parameter tractable (FPT) algorithm in $k$ and $m$, that almost matches the approximation ratio for its outlier-free counterpart. As a corollary, we obtain FPT approximation algorithms with o…
Given a dataset $V$ of points from some metric space, the popular $k$-center problem requires to identify a subset of $k$ points (centers) in $V$ minimizing the maximum distance of any point of $V$ from its closest center. The \emph{robust} formulation of the problem features a further parameter $z$ and allows up to $z$ points of $V$ (outliers) to be disregarded when computing the maximum distance from the centers. In this paper, we focus on two important constrained variants of the robust $k$-center problem, namely, the Robust Matroid Center (RMC) problem, where the set of returned centers ar…
In the Priority $k$-Center problem, the input consists of a metric space $(X,d)$, an integer $k$, and for each point $v \in X$ a priority radius $r(v)$. The goal is to choose $k$-centers $S \subseteq X$ to minimize $\max_{v \in X} \frac{1}{r(v)} d(v,S)$. If all $r(v)$'s are uniform, one obtains the $k$-Center problem. Plesník [Plesník, Disc. Appl. Math. 1987] introduced the Priority $k$-Center problem and gave a $2$-approximation algorithm matching the best possible algorithm for $k$-Center. We show how the problem is related to two different notions of fair clustering [Harris et al., NeurIPS …
Constant-Factor Approximations for Doubly Constrained Fair k-Center, k-Median and k-Means
Department of Informatics, University of Cologne, Germany（科隆大学信息学院，德国）；Heinrich Heine University Düsseldorf, Faculty of Mathematics and Natural Sciences, Germany（杜伊斯堡-艾森大学数学与自然科学学院，德国）；Department of Informatics, University of Bonn, Germany（波恩大学信息学院，德国）
In this paper, we give first constant factor approximation for capacitated knapsack median problem (CKM) for hard uniform capacities, violating the budget only by an additive factor of $f_{max}$ where $f_{max}$ is the maximum cost of a facility opened by the optimal and violating capacities by $(2+ε)$ factor. Natural LP for the problem is known to have an unbounded integrality gap when any one of the two constraints is allowed to be violated by a factor less than $2$. Thus, we present a result which is very close to the best achievable from the natural LP. To the best of our knowledge, the pro…
