Ranking systems based on pairwise comparisons are fundamental in decision-making applications, yet fairness concerns remain largely unaddressed, particularly in active ranking frameworks. We propose a novel fairness-aware active ranking approach that adaptively queries pairwise preferences to construct rankings that are both probably approximately correct (PAC) and fair. We propose a flexible cascaded norm-based objective that balances error distribution within and across socially salient groups, providing a unified framework to mitigate systemic disparities. Adopting our objective function allows us to explore fundamental fairness concepts like equal or proportionate errors within a unified framework. We develop both group-blind and group-aware algorithms and derive their sample complexity bounds. Empirical evaluations on real-world datasets, including COMPAS and German Credit, demonstrate the efficiency of our approach, reducing sample complexity while achieving fairer rankings. Our findings offer theoretical insights and practical methods to enhance fairness in active ranking systems, improving their reliability in hiring, recommendations, and other applications.

This work studies a novel subset selection problem called *max-min diversification with monotone submodular utility* (MDMS), which has a wide range of applications in machine learning, e.g., data sampling and feature selection. Given a set of points in a metric space, the goal of MDMS is to maximize $f(S) = g(S) + \lambda \cdot \text{div}(S)$ subject to a cardinality constraint $|S| \le k$, where $g(S)$ is a monotone submodular function and $\text{div}(S) = \min_{u,v \in S : u \ne v} \text{dist}(u,v)$ is the *max-min diversity* objective. We propose the `GIST` algorithm, which gives a $\frac{1}{2}$-approximation guarantee for MDMS by approximating a series of maximum independent set problems with a bicriteria greedy algorithm. We also prove that it is NP-hard to approximate within a factor of $0.5584$. Finally, we show in our empirical study that `GIST` outperforms state-of-the-art benchmarks for a single-shot data sampling task on ImageNet.

The classic Mallows model is a foundational tool for modeling user preferences. However, it has limitations in capturing real-world scenarios, where users often focus only on a limited set of preferred items and are indifferent to the rest. To address this, extensions such as the top-$k$ Mallows model have been proposed, aligning better with practical applications. In this paper, we address several challenges related to the generalized top-$k$ Mallows model, with a focus on analyzing buyer choices. Our key contributions are: (1) a novel sampling scheme tailored to generalized top-$k$ Mallows models, (2) an efficient algorithm for computing choice probabilities under this model, and (3) an active learning algorithm for estimating the model parameters from observed choice data. These contributions provide new tools for analysis and prediction in critical decision-making scenarios. We present a rigorous mathematical analysis for the performance of our algorithms. Furthermore, through extensive experiments on synthetic data and real-world data, we demonstrate the scalability and accuracy of our proposed methods, and we compare the predictive power of Mallows model for top-$k$ lists compared to the simpler Multinomial Logit model.

We study the problem of approximately counting cliques and near-cliques in a graph, where the access to the graph is only available through crawling its vertices. This model has been introduced recently to capture real-life scenarios in which the entire graph is too massive to be stored as a whole or be scanned entirely. Sampling vertices independently is non-trivial in this model, thus algorithms which rely on sampling often use a random walk. The goal is to provide an accurate estimate by seeing only a small portion of the graph. This model is known as the random walk model or the neighborhood query model. We introduce DeMEtRIS : Dense Motif Estimation through Random Incident Sampling. This method provides a scalable algorithm for clique and near-clique counting in the random walk model. We prove the correctness of our algorithm through rigorous mathematical analysis and extensive experiments. Both our theoretical results and our experiments show that DeMEtRIS obtains a high precision estimation by only crawling a sub-linear portion on vertices. Therefore, we demonstrate a significant improvement over previous known results.

Cardinality sketches are popular data structures that enhance the efficiency of working with large data sets. The sketches are randomized representations of sets that are only of logarithmic size but can support set merges and approximate cardinality (i.e., distinct count) queries. When queries are not adaptive, that is, they do not depend on preceding query responses, the design provides strong guarantees of correctly answering a number of queries exponential in the sketch size $k$. In this work, we investigate the performance of cardinality sketches in adaptive settings and unveil inherent vulnerabilities. We design an attack against the “standard” estimators that constructs an adversarial input by post-processing responses to a set of simple non-adaptive queries of size linear in the sketch size $k$. Empirically, our attack used only $4k$ queries with the widely used HyperLogLog (HLL++) Flajolet et al., 2007; Heule et al., 2013) sketch. The simple attack technique suggests it can be effective with post-processed natural workloads. Finally and importantly, we demonstrate that the vulnerability is inherent as any estimator applied to known sketch structures can be attacked using a number of queries that is quadratic in $k$, matching a generic upper bound.

We study the problem of resilient clustering in the metric setting where one is interested in designing algorithms that return high quality solutions that preserve the clustering structure under perturbations of the input points. Our first contribution is to introduce a formal notion of algorithmic resiliency for clustering problems that, roughly speaking, requires an algorithm to have similar outputs on close inputs. Then, we notice that classic algorithms have weak resiliency guarantees and develop new algorithms for fundamental clustering problems such as k-center, k-median, and k-means. Finally, we complement our results with an experimental analysis showing the effectiveness of our techniques on real-world instances.

Online navigation platforms are well optimized to solve the standard objective of minimizing travel time and typically require precomputation-based architectures (such as Contraction Hierarchies and Customizable Route Planning) to do so in a fast manner. The reason for this dependence is the size of the graph that represents the road network, which is large. The need to go beyond minimizing the travel time and introduce various types of customizations has led to approaches that rely on alternative route computation or, more generally, small subgraph extraction. On a small subgraph, one can run computationally expensive algorithms at query time and compute optimal solutions for multiple routing problems. In this framework, it is critical for the subgraph to (a) be small and (b) include (near) optimal routes for a collection of customizations. This is precisely the setting that we study in this work. We design algorithms that extract a subgraph connecting designated terminals with the objective of minimizing the subgraph's size and the constraint of including near-optimal routes for a set of predefined cost functions. We provide theoretical guarantees for our algorithms and evaluate them empirically using real-world road networks.

Correlation clustering is a ubiquitous paradigm in unsupervised machine learning where addressing unfairness is a major challenge. Motivated by this, we study fair correlation clustering where the data points may belong to different protected groups and the goal is to ensure fair representation of all groups across clusters. Our paper significantly generalizes and improves on the quality guarantees of previous work of Ahmadian et al. as follows. * We allow the user to specify an arbitrary upper bound on the representation of each group in a cluster. * Our algorithm allows individuals to have multiple protected features and ensure fairness simultaneously across them all. * We prove guarantees for clustering quality and fairness in this general setting. Furthermore, this improves on the results for the special cases studied in previous work. Our experiments on real-world data demonstrate that our clustering quality compared to the optimal solution is much better than what our theoretical result suggests.

We study the problem of approximately counting cliques and near cliques in a graph, where the access to the graph is only available through crawling its vertices; thus typically seeing only a small portion of it. This model, known as the random walk model or the neighborhood query model has been introduced recently and captures real-life scenarios in which the entire graph is too massive to be stored as a whole or be scanned entirely and sampling vertices independently is non-trivial in it.

Motivated by the exploding growth of web-based services and the importance of efficiently managing the computational resources of such systems, we introduce and study a theoretical model for load balancing of very large databases such as commercial search engines. Our model is a more realistic version of the well-received \bab model with an additional constraint that limits the number of servers that carry each piece of the data. This additional constraint is necessary when, on one hand, the data is so large that we can not copy the whole data on each server. On the other hand, the query response time is so limited that we can not ignore the fact that the number of queries for each piece of the data changes over time, and hence we can not simply split the data over different machines. In this paper, we develop an almost optimal load balancing algorithm that works given an estimate of the load of each piece of the data. Our algorithm is almost perfectly robust to wrong estimates, to the extent that even when all of the loads are adversarially chosen the performance of our algorithm is $1-1/e$, which is provably optimal. Along the way, we develop various techniques for analyzing the balls-into-bins process under certain correlations and build a novel connection with the multiplicative weights update scheme.

Motivated by the exploding growth of web-based services and the importance of efficiently managing the computational resources of such systems, we introduce and study a theoretical model for load balancing of very large databases such as commercial search engines. Our model is a more realistic version of the well-received balls-into-bins model with an additional constraint that limits the number of servers that carry each piece of the data. This additional constraint is necessary when, on one hand, the data is so large that we can not copy the whole data on each server. On the other hand, the query response time is so limited that we can not ignore the fact that the number of queries for each piece of the data changes over time, and hence we can not simply split the data over different machines In this paper, we develop an almost optimal load balancing algorithm that works given an estimate of the load of each piece of the data. Our algorithm is almost perfectly robust to wrong estimates, to the extent that even when all of the loads are adversarially chosen the performance of our algorithm is 1–1/e, which is provably optimal. Along the way, we develop various techniques for analyzing the balls-into-bins process under certain correlations and build a novel connection with the multiplicative weights update scheme.

In this paper, we study a number of well-known combinatorial optimization problems that fit in the following paradigm: the input is a collection of (potentially inconsistent) local relationships between the elements of a ground set (e.g., pairwise comparisons, similar/dissimilar pairs, or ancestry structure of triples of points), and the goal is to aggregate this information into a global structure (e.g., a ranking, a clustering, or a hierarchical clustering) in a way that maximizes agreement with the input. Well-studied problems such as rank aggregation, correlation clustering, and hierarchical clustering with triplet constraints fall in this class of problems. We study these problems on stochastic instances with a hidden embedded ground truth solution. Our main algorithmic contribution is a unified technique that uses the maximum cut problem in graphs to approximately solve these problems. Using this technique, we can often get approximation guarantees in the stochastic setting that are better than the known worst case inapproximability bounds for the corresponding problem. On the negative side, we improve the worst case inapproximability bound on several hierarchical clustering formulations through a reduction to related ranking problems.

Inspired by applications on search engines and web servers, we consider a load balancing problem with a general convex objective function. In this problem, we are given a bipartite graph on a set of sources S and a set of workers W and the goal is to distribute the load from each source among its neighboring workers such that the total load of workers are as balanced as possible. We present a new distributed algorithm that works with any symmetric non-decreasing convex function for evaluating the balancedness of the workers' load. Our algorithm computes a nearly optimal allocation of loads in O(log n log² d/ε³) rounds where n is the number of nodes, d is the maximum degree, and ε is the desired precision. If the objective is to minimize the maximum load, we modify the algorithm to obtain a nearly optimal solution in O(log n log d/ε²) rounds. This improves a line of algorithms that require a polynomial number of rounds in n and d and appear to encounter a fundamental barrier that prevents them from obtaining poly-logarithmic runtime [Berenbrink et al., 2005; Berenbrink et al., 2009; Subramanian and Scherson, 1994; Rabani et al., 1998]. In our paper, we introduce a novel primal-dual approach with multiplicative weight updates that allows us to circumvent this barrier. Our algorithm is inspired by [Agrawal et al., 2018] and other distributed algorithms for optimizing linear objectives but introduces several new twists to deal with general convex objectives.

In this paper, we study correlation clustering under fairness constraints. Fair variants of $k$-median and $k$-center clustering have been studied recently, and approximation algorithms using a notion called fairlet decomposition have been proposed. We obtain approximation algorithms for fair correlation clustering under several important types of fairness constraints.Our results hinge on obtaining a fairlet decomposition for correlation clustering by introducing a novel combinatorial optimization problem. We define a fairlet decomposition with cost similar to the $k$-median cost and this allows us to obtain approximation algorithms for a wide range of fairness constraints. We complement our theoretical results with an in-depth analysis of our algorithms on real graphs where we show that fair solutions to correlation clustering can be obtained with limited increase in cost compared to the state-of-the-art (unfair) algorithms.

Hierarchical Clustering is an unsupervised data analysis method which has been widely used for decades. Despite its popularity, it had an underdeveloped analytical foundation and to address this, Dasgupta recently introduced an optimization viewpoint of hierarchical clustering with pairwise similarity information that spurred a line of work shedding light on old algorithms (e.g., Average-Linkage), but also designing new algorithms. Here, for the maximization dual of Dasgupta’s objective (introduced by Moseley-Wang), we present polynomial-time 42.46% approximation algorithms that use Max-Uncut Bisection as a subroutine. The previous best worst-case approximation factor in polynomial time was 33.6%, improving only slightly over Average-Linkage which achieves 33.3%. Finally, we complement our positive results by providing APX-hardness (even for 0-1 similarities), under the Small Set Expansion hypothesis.

Social networks have become an inseparable part of human life and processing them in an efficient manner is a top priority in the study of networks. These networks are highly dynamic and they are growing incessantly. Inspired by the concept of triadic closure, we propose a probabilistic mechanism to model the evolution of these dynamic graphs. Although triadic closure is ubiquitous in social networks and its presence helps forming communities, probabilistic models encapsulating it have not been studied adequately.We theoretically analyze our model and show how to bound the growth rate of some characteristics of the graph, such as degree of vertices. Leveraging our theoretical results, we develop a scheduling subroutine to process modifications o f t he graph in batches. Our scheduling subroutine is then used to speed up the state-of-the-art algorithms with negligible loss in their approximation guarantees. We demonstrate the applicability of our method by applying it to the densest subgraph discovery problem.

As machine learning has become more prevalent, researchers have begun to recognize the necessity of ensuring machine learning systems are fair. Recently, there has been an interest in defining a notion of fairness that mitigates over-representation in traditional clustering. In this paper we extend this notion to hierarchical clustering, where the goal is to recursively partition the data to optimize a specific objective. For various natural objectives, we obtain simple, efficient algorithms to find a provably good fair hierarchical clustering. Empirically, we show that our algorithms can find a fair hierarchical clustering, with only a negligible loss in the objective.

Clustering is a classic topic in optimization with $k$-means being one of the most fundamental such problems. In the absence of any restrictions on the input, the best-known algorithm for $k$-means in Euclidean space with a provable guarantee is a simple local search heuristic yielding an approximation guarantee of $9+\epsilon$, a ratio that is known to be tight with respect to such methods. We overcome this barrier by presenting a new primal-dual approach that allows us to (1) exploit the geometric structure of $k$-means and (2) satisfy the hard constraint that at most $k$ clusters are selected without deteriorating the approximation guarantee. Our main result is a 6.357-approximation algorithm with respect to the standard linear programming (LP) relaxation. Our techniques are quite general, and we also show improved guarantees for $k$-median in Euclidean metrics and for a generalization of $k$-means in which the underlying metric is not required to be Euclidean.

In this paper we consider clustering problems in which each point is endowed with a color. The goal is to cluster the points to minimize the classical clustering cost but with the additional constraint that no color is over-represented in any cluster. This problem is motivated by practical clustering settings, e.g., in clustering news articles where the color of an article is its source, it is preferable that no single news source dominates any cluster. For the most general version of this problem, we obtain an algorithm that has provable guarantees of performance; our algorithm is based on finding a fractional solution using a linear program and rounding the solution subsequently. For the special case of the problem where no color has an absolute majority in any cluster, we obtain a simpler combinatorial algorithm also with provable guarantees. Experiments on real-world data shows that our algorithms are effective in finding good clustering without over-representation.

We study inverse optimization problems, wherein the goal is to map given solutions to an underlying optimization problem to a cost vector for which the given solutions are the (unique) optimal solutions. Inverse optimization problems find diverse applications and have been widely studied. A prominent problem in this field is the inverse shortest path (ISP) problem [D. Burton and Ph.L. Toint, 1992; W. Ben-Ameur and E. Gourdin, 2004; A. Bley, 2007], which finds applications in shortest-path routing protocols used in telecommunications. Here we seek a cost vector that is positive, integral, induces a set of given paths as the unique shortest paths, and has minimum l_infty norm. Despite being extensively studied, very few algorithmic results are known for inverse optimization problems involving integrality constraints on the desired cost vector whose norm has to be minimized. Motivated by ISP, we initiate a systematic study of such integral inverse optimization problems from the perspective of designing polynomial time approximation algorithms. For ISP, our main result is an additive 1-approximation algorithm for multicommodity ISP with node-disjoint commodities, which we show is tight assuming P!=NP. We then consider the integral-cost inverse versions of various other fundamental combinatorial optimization problems, including min-cost flow, max/min-cost bipartite matching, and max/min-cost basis in a matroid, and obtain tight or nearly-tight approximation guarantees for these. Our guarantees for the first two problems are based on results for a broad generalization, namely integral inverse polyhedral optimization, for which we also give approximation guarantees. Our techniques also give similar results for variants, including l_p-norm minimization of the integral cost vector, and distance-minimization from an initial cost vector.

We consider a facility-location problem that abstracts settings where the cost of serving the clients assigned to a facility is incurred by the facility. Formally, we consider the minimum-load k-facility location (ML k FL) problem, which is defined as follows. We have a set F of facilities, a set C of clients, and an integer k ≥ 0. Assigning client j to a facility f incurs a connection cost d ( f , j ). The goal is to open a set F ⊆ F of k facilities and assign each client j to a facility f ( j )∈ F so as to minimize max f ∈ F ∑ j ∈ C : f ( j )= f d ( f , j ); we call ∑ j ∈ C : f ( j )= f d ( f , j ) the load of facility f . This problem was studied under the name of min-max star cover in References [3, 7], who (among other results) gave bicriteria approximation algorithms for ML k FL for when F = C . ML k FL is rather poorly understood, and only an O ( k )-approximation is currently known for ML k FL, even for line metrics . Our main result is the first polytime approximation scheme (PTAS) for ML k FL on line metrics (note that no non-trivial true approximation of any kind was known for this metric). Complementing this, we prove that ML k FL is strongly NP -hard on line metrics. We also devise a quasi-PTAS for ML k FL on tree metrics. ML k FL turns out to be surprisingly challenging even on line metrics and resilient to attack by a variety of techniques that have been successfully applied to facility-location problems. For instance, we show that (a) even a configuration-style LP-relaxation has a bad integrality gap and (b) a multi-swap k -median style local-search heuristic has a bad locality gap. Thus, we need to devise various novel techniques to attack ML k FL. Our PTAS for line metrics consists of two main ingredients. First, we prove that there always exists a near-optimal solution possessing some nice structural properties. A novel aspect of this proof is that we first move to a mixed-integer LP (MILP) encoding of the problem and argue that a MILP-solution minimizing a certain potential function possesses the desired structure and then use a rounding algorithm for the generalized-assignment problem to “transfer” this structure to the rounded integer solution. Complementing this, we show that these structural properties enable one to find such a structured solution via dynamic programming.

Clustering is a classic topic in optimization with k-means being one of the most fundamental such problems. In the absence of any restrictions on the input, the best known algorithm for k-means with a provable guarantee is a simple local search heuristic yielding an approximation guarantee of 9 + ε, a ratio that is known to be tight with respect to such methods. We overcome this barrier by presenting a new primal-dual approach that allows us to (1) exploit the geometric structure of k-means and (2) to satisfy the hard constraint that at most k clusters are selected without deteriorating the approximation guarantee. Our main result is a 6.357-approximation algorithm with respect to the standard LP relaxation. Our techniques are quite general and we also show improved guarantees for the general version of k-means where the underlying metric is not required to be Euclidean and for k-median in Euclidean metrics.

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 the vertex capacities. This problem is APX-hard and constant-factor approximations are already known. Our main results fall into two categories. First, using iterated relaxation and various filtering strategies, we show with an efficient rounding algorithm that if an additional matroid structure M is given and we further only allow sets that are independent in M, the natural LP relaxation has an integrality gap of at most 25/3. This can be further improved in various special cases, for example we improve over the 15-approximation for the previously- studied Coupled Placement problem [Korupolu et al. 2014] by giving a 7-approximation. Using similar techniques, we show the problem of computing a minimum-cost base in M satisfying vertex capacities admits a (1,3)-bicriteria approximation: the cost is at most the optimum and the capacities are violated by a factor of at most 3. This improves over the previous (1,4)-approximation in the special case that M is the graphic matroid over the given graph [Fukanaga and Nagamochi, 2009]. Second, we show Demand Matching admits a polynomial-time approximation scheme in graphs that exclude a fixed minor. If all demands are polynomially-bounded integers, this is somewhat easy using dynamic programming in bounded-treewidth graphs. Our main technical contribution is a sparsification lemma that allows us to scale the demands of some items to be used in a more intricate dynamic programming algorithm, followed by some randomized rounding to filter our scaled-demand solution to one whose original demands satisfy all constraints.

We consider clustering problems with non-uniform lower bounds and outliers, and obtain the first approximation guarantees for these problems. We have a set F of facilities with lower bounds {L_i}_{i in F} and a set D of clients located in a common metric space {c(i,j)}_{i,j in F union D}, and bounds k, m. A feasible solution is a pair (S subseteq F, sigma: D -> S union {out}), where sigma specifies the client assignments, such that |S| <=k, |sigma^{-1}(i)| >= L_i for all i in S, and |sigma^{-1}(out)| <= m. In the lower-bounded min-sum-of-radii with outliers P (LBkSRO) problem, the objective is to minimize sum_{i in S} max_{j in sigma^{-1})i)}, and in the lower-bounded k-supplier with outliers (LBkSupO) problem, the objective is to minimize max_{i in S} max_{j in sigma^{-1})i)} c(i,j). We obtain an approximation factor of 12.365 for LBkSRO, which improves to 3.83 for the non-outlier version (i.e., m = 0). These also constitute the first approximation bounds for the min-sum-of-radii objective when we consider lower bounds and outliers separately. We apply the primal-dual method to the relaxation where we Lagrangify the |S| <= k constraint. The chief technical contribution and novelty of our algorithm is that, departing from the standard paradigm used for such constrained problems, we obtain an O(1)-approximation despite the fact that we do not obtain a Lagrangian-multiplier-preserving algorithm for the Lagrangian relaxation. We believe that our ideas have broader applicability to other clustering problems with outliers as well. We obtain approximation factors of 5 and 3 respectively for LBkSupO and its non-outlier version. These are the first approximation results for k-supplier with non-uniform lower bounds.

We consider a facility-location problem that abstracts settings where the cost of serving the clients assigned to a facility is incurred by the facility. Formally, we consider the minimum-load k-facility location (MLkFL) problem, which is defined as follows. We have a set F of facilities, a set C of clients, and an integer k > 0. Assigning client j to a facility f incurs a connection cost d(f, j). The goal is to open a set F' of k facilities, and assign each client j to a facility f(j) in F' so as to minimize maximum, over all facilities in F', of the sum of distances of clients j assigned to F' to F'. We call this sum the load of facility f. This problem was studied under the name of min-max star cover in [6, 2], who (among other results) gave bicriteria approximation algorithms for MLkFL for when F = C. MLkFL is rather poorly understood, and only an O(k)-approximation is currently known for MLkFL, even for line metrics. Our main result is the first polynomial time approximation scheme (PTAS) for MLkFL on line metrics (note that no non-trivial true approximation of any kind was known for this metric). Complementing this, we prove that MLkFL is strongly NP-hard on line metrics. We also devise a quasi-PTAS for MLkFL on tree metrics. MLkFL turns out to be surprisingly challenging even on line metrics, and resilient to attack by the variety of techniques that have been successfully applied to facility-location problems. For instance, we show that: (a) even a configuration-style LP-relaxation has a bad integrality gap; and (b) a multi-swap k-median style local-search heuristic has a bad locality gap. Thus, we need to devise various novel techniques to attack MLkFL. Our PTAS for line metrics consists of two main ingredients. First, we prove that there always exists a near-optimal solution possessing some nice structural properties. A novel aspect of this proof is that we first move to a mixed-integer LP (MILP) encoding the problem, and argue that a MILP-solution minimizing a certain potential function possesses the desired structure, and then use a rounding algorithm for the generalized-assignment problem to "transfer" this structure to the rounded integer solution. Complementing this, we show that these structural properties enable one to find such a structured solution via dynamic programming.

We consider the mobile facility location (MFL) problem. We are given a set of facilities and clients located in a common metric space G = (V, c). The goal is to move each facility from its initial location to a destination (in V) and assign each client to the destination of some facility so as to minimize the sum of the movement-costs of the facilities and the client-assignment costs. This abstracts facility-location settings where one has the flexibility of moving facilities from their current locations to other destinations so as to serve clients more efficiently by reducing their assignment costs. We give the first local-search based approximation algorithm for this problem and achieve the best-known approximation guarantee. Our main result is (3 + ε)-approximation for this problem for any constant ε > 0 using local search. The previous best guarantee for MFL was an 8-approximation algorithm due to Friggstad and Salavatipour [12] based on LP-rounding. Our guarantee matches the best-known approximation guarantee for the k-median problem. Since there is an approximation-preserving reduction from the k-median problem to MFL, any improvement of our result would imply an analogous improvement for the k-median problem. Furthermore, our analysis is tight (up to o(1) factors) since the tight example for the local-search based 3-approximation algorithm for k-median can be easily adapted to show that our local-search algorithm has a tight approximation ratio of 3. Our results extend to the weighted generalization wherein each facility i has a non-negative weight wi and the movement cost for i is wi times the distance traveled by i. In contrast to the k-median problem, the local search procedure that moves, at each step, a constant number of facilities (to chosen destinations) and assigns each client to the nearest destination, is known to have an unbounded locality gap. Our local-search algorithm is a natural and simple variant, where we only select the destinations of the facilities in each step and optimally rematch the facilities to these destinations (which might entail moving all facilities). One of the chief novelties in the analysis is that in order to generate a suitable collection of local-search moves whose resulting inequalities yield the desired bound on the cost of a local-optimum, we define a tree-like structure that (loosely speaking) functions as a “recursion tree”, using which we spawn off local-search moves by exploring this tree to a constant depth.