We present a new local-search algorithm for the $k$-median clustering problem. We show that local optima for this algorithm give a $(2.836+ε)$-approximation; our result improves upon the $(3+ε)$-approximate local-search algorithm of Arya et al. [STOC 01]. Moreover, a computer-aided analysis of a natural extension suggests that this approach may lead to an improvement over the best-known approximation guarantee for the problem. The new ingredient in our algorithm is the use of a potential function based on both the closest and second-closest facilities to each client. Specifically, the potential is the sum over all clients, of the distance of the client to its closest facility, plus (a small constant times) the truncated distance to its second-closest facility. We move from one solution to another only if the latter can be obtained by swapping a constant number of facilities, and has a smaller potential than the former. This refined potential allows us to avoid the bad local optima given by Arya et al. for the local-search algorithm based only on the cost of the solution.
The k-median problem is a well-known strongly NP-hard combinatorial optimization problem of both theoretical and practical significance. The previous best approximation ratio for this problem is 2.611+ε(Bryka et al. 2014) based on an (1, 1.95238219) bi-factor approximation algorithm for the classical facility location problem (FLP). This work offers an improved algorithm with an approximation ratio 2.592 +εbased on a new (1, 1.93910094) bi-factor approximation algorithm for the FLP.
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…
We extend randomized jumplists introduced by Brönnimann et al. (STACS 2003) to choose jump-pointer targets as median of a small sample for better search costs, and present randomized algorithms with expected $O(\log n)$ time complexity that maintain the probability distribution of jump pointers upon insertions and deletions. We analyze the expected costs to search, insert and delete a random element, and we show that omitting jump pointers in small sublists hardly affects search costs, but significantly reduces the memory consumption. We use a bijection between jumplists and "dangling-min BSTs…
Dependent rounding is a useful technique for optimization problems with hard budget constraints. This framework naturally leads to \emph{negative correlation} properties. However, what if an application naturally calls for dependent rounding on the one hand, and desires \emph{positive} correlation on the other? More generally, we develop algorithms that guarantee the known properties of dependent rounding, but also have nearly best-possible behavior - near-independence, which generalizes positive correlation - on "small" subsets of the variables. The recent breakthrough of Li & Svensson for th…
The current best approximation algorithms for $k$-median rely on first obtaining a structured fractional solution known as a bi-point solution, and then rounding it to an integer solution. We improve this second step by unifying and refining previous approaches. We describe a hierarchy of increasingly-complex partitioning schemes for the facilities, along with corresponding sets of algorithms and factor-revealing non-linear programs. We prove that the third layer of this hierarchy is a $2.613$-approximation, improving upon the current best ratio of $2.675$, while no layer can be proved better …
In this paper, we study the hard uniform capacitated $k$- median problem using local search heuristic. Obtaining a constant factor approximation for the \ckm problem is open. All the existing solutions giving constant-factor approximation, violate at least one of the cardinality and the capacity constraints. All except Koruplou et al are based on LP-relaxation. We give $(3+ε)$ factor approximation algorithm for the problem violating the cardinality by a factor of $8/3 \approx 2.67$. There is a trade-off between the approximation factor and the cardinality violation between our work and the exi…
A common problem appearing across the field of data science is $k$-NN ($k$-nearest neighbours), particularly within the context of Geographic Information Systems. In this article, we present a novel data structure, the GeoTree, which holds a collection of geohashes (string encodings of GPS co-ordinates). This enables a constant $O\left(1\right)$ time search algorithm that returns a set of geohashes surrounding a given geohash in the GeoTree, representing the approximate $k$-nearest neighbours of that geohash. Furthermore, the GeoTree data structure retains $O\left(n\right)$ memory requirement.…
We give the first polynomial-time approximation schemes (PTASs) for the following problems: (1) uniform facility location in edge-weighted planar graphs; (2) $k$-median and $k$-means in edge-weighted planar graphs; (3) $k$-means in Euclidean spaces of bounded dimension. Our first and second results extend to minor-closed families of graphs. All our results extend to cost functions that are the $p$-th power of the shortest-path distance. The algorithm is local search where the local neighborhood of a solution $S$ consists of all solutions obtained from $S$ by removing and adding $1/ε^{O(1)}$ ce…
Consider a generalization of the classical binary search problem in linearly sorted data to the graph-theoretic setting. The goal is to design an adaptive query algorithm, called a strategy, that identifies an initially unknown target vertex in a graph by asking queries. Each query is conducted as follows: the strategy selects a vertex $q$ and receives a reply $v$: if $q$ is the target, then $v=q$, and if $q$ is not the target, then $v$ is a neighbor of $q$ that lies on a shortest path to the target. Furthermore, there is a noise parameter $0\leq p<\frac{1}{2}$, which means that each reply can…
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…
Google Research（谷歌研究院）；Massachusetts Institute of Technology（麻省理工学院）
Motivated by data analysis and machine learning applications, we consider the popular high-dimensional Euclidean $k$-median and $k$-means problems. We propose a new primal-dual algorithm, inspired by the classic algorithm of Jain and Vazirani and the recent algorithm of Ahmadian, Norouzi-Fard, Svensson, and Ward. Our algorithm achieves an approximation ratio of $2.406$ and $5.912$ for Euclidean $k$-median and $k$-means, respectively, improving upon the 2.633 approximation ratio of Ahmadian et al. and the 6.1291 approximation ratio of Grandoni, Ostrovsky, Rabani, Schulman, and Venkat. Our techn…
In the classical binary search in a path the aim is to detect an unknown target by asking as few queries as possible, where each query reveals the direction to the target. This binary search algorithm has been recently extended by [Emamjomeh-Zadeh et al., STOC, 2016] to the problem of detecting a target in an arbitrary graph. Similarly to the classical case in the path, the algorithm of Emamjomeh-Zadeh et al. maintains a candidates' set for the target, while each query asks an appropriately chosen vertex-- the "median"--which minimises a potential $Φ$ among the vertices of the candidates' set.…
When designing clustering algorithms, the choice of initial centers is crucial for the quality of the learned clusters. In this paper, we develop a new initialization scheme, called HST initialization, for the $k$-median problem in the general metric space (e.g., discrete space induced by graphs), based on the construction of metric embedding tree structure of the data. From the tree, we propose a novel and efficient search algorithm, for good initial centers that can be used subsequently for the local search algorithm. Our proposed HST initialization can produce initial centers achieving lowe…
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 deterioratin…
University at Buffalo（纽约州立大学布法罗分校）；Microsoft Research（微软研究院）
In this paper we introduce and study the online consistent $k$-clustering with outliers problem, generalizing the non-outlier version of the problem studied in [Lattanzi-Vassilvitskii, ICML17]. We show that a simple local-search based online algorithm can give a bicriteria constant approximation for the problem with $O(k^2 \log^2 (nD))$ swaps of medians (recourse) in total, where $D$ is the diameter of the metric. When restricted to the problem without outliers, our algorithm is simpler, deterministic and gives better approximation ratio and recourse, compared to that of [Lattanzi-Vassilvitski…
The $k$-Facility Location problem is a generalization of the classical problems $k$-Median and Facility Location. The goal is to select a subset of at most $k$ facilities that minimizes the total cost of opened facilities and established connections between clients and opened facilities. We consider the hard-capacitated version of the problem, where a single facility may only serve a limited number of clients and creating multiple copies of a facility is not allowed. We construct approximation algorithms slightly violating the capacities based on rounding a fractional solution to the standard …
Clustering problems are well-studied in a variety of fields such as data science, operations research, and computer science. Such problems include variants of centre location problems, $k$-median, and $k$-means to name a few. In some cases, not all data points need to be clustered; some may be discarded for various reasons. We study clustering problems with outliers. More specifically, we look at Uncapacitated Facility Location (UFL), $k$-Median, and $k$-Means. In UFL with outliers, we have to open some centres, discard up to $z$ points of $\cal X$ and assign every other point to the nearest o…
We study the Ordered k-Median problem, in which the solution is evaluated by first sorting the client connection costs and then multiplying them with a predefined non-increasing weight vector (higher connection costs are taken with larger weights). Since the 1990s, this problem has been studied extensively in the discrete optimization and operations research communities and has emerged as a framework unifying many fundamental clustering and location problems such as k-Median and k-Center. This generality, however, renders the problem intriguing from the algorithmic perspective and obtaining no…
The median of a set of vertices $P$ of a graph $G$ is the set of all vertices $x$ of $G$ minimizing the sum of distances from $x$ to all vertices of $P$. In this paper, we present a linear time algorithm to compute medians in median graphs, improving over the existing quadratic time algorithm. We also present a linear time algorithm to compute medians in the $\ell_1$-cube complexes associated with median graphs. Median graphs constitute the principal class of graphs investigated in metric graph theory and have a rich geometric and combinatorial structure, due to their bijections with CAT(0) cu…
In the k-median problem we are given sets of facilities and customers, and distances between them. For a given set F of facilities, the cost of serving a customer u is the minimum distance between u and a facility in F. The goal is to find a set F of k facilities that minimizes the sum, over all customers, of their service costs. Following Mettu and Plaxton, we study the incremental medians problem, where k is not known in advance, and the algorithm produces a nested sequence of facility sets where the kth set has size k. The algorithm is c-cost-competitive if the cost of each set is at most c…
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…
What is the effectiveness of local search algorithms for geometric problems in the plane? We prove that local search with neighborhoods of magnitude $1/ε^c$ is an approximation scheme for the following problems in the Euclidian plane: TSP with random inputs, Steiner tree with random inputs, facility location (with worst case inputs), and bicriteria $k$-median (also with worst case inputs). The randomness assumption is necessary for TSP.
We study the k-median and k-center problems in probabilistic graphs. We analyze the hardness of these problems, and propose several algorithms with improved approximation ratios compared with the existing proposals.
In the time-decay model for data streams, elements of an underlying data set arrive sequentially with the recently arrived elements being more important. A common approach for handling large data sets is to maintain a \emph{coreset}, a succinct summary of the processed data that allows approximate recovery of a predetermined query. We provide a general framework that takes any offline-coreset and gives a time-decay coreset for polynomial time decay functions. We also consider the exponential time decay model for $k$-median clustering, where we provide a constant factor approximation algorithm …
We propose a new variant of the k-median problem, where the objective function models not only the cost of assigning data points to cluster representatives, but also a penalty term for disagreement among the representatives. We motivate this novel problem by applications where we are interested in clustering data while avoiding selecting representatives that are too far from each other. For example, we may want to summarize a set of news sources, but avoid selecting ideologically-extreme articles in order to reduce polarization. To solve the proposed k-median formulation we adopt the local-sea…
Google Research（谷歌研究院）；Aarhus University（奥胡斯大学）；Sorbonne Université（索邦大学）
Given a set of points in a metric space, the $(k,z)$-clustering problem consists of finding a set of $k$ points called centers, such that the sum of distances raised to the power of $z$ of every data point to its closest center is minimized. Special cases include the famous k-median problem ($z = 1$) and k-means problem ($z = 2$). The $k$-median and $k$-means problems are at the heart of modern data analysis and massive data applications have given raise to the notion of coreset: a small (weighted) subset of the input point set preserving the cost of any solution to the problem up to a multipl…
In this paper, we discuss an improvement of an algorithm to search for primes $p$ and coset-partitions of Z/pZ* that yield Ramsey algebras over Z/pZ. We also prove an upper bound on the modulus p in terms of the number of cosets. We have, as a corollary, that there is no prime $p$ for which there exists a partition of Z/pZ* into 13 cosets that yields a 13-color Ramsey algebra. Thus A263308(13) = 0.
The edit distance under the DCJ model can be computed in linear time for genomes with equal content or with Indels. But it becomes NP-Hard in the presence of duplications, a problem largely unsolved especially when Indels are considered. In this paper, we compare two mainstream methods to deal with duplications and associate them with Indels: one by deletion, namely DCJ-Indel-Exemplar distance; versus the other by gene matching, namely DCJ-Indel-Matching distance. We design branch-and-bound algorithms with set of optimization methods to compute exact distances for both. Furthermore, median pro…
Recently it was shown that the problem of Maximum Inner Product Search (MIPS) is efficient and it admits provably sub-linear hashing algorithms. Asymmetric transformations before hashing were the key in solving MIPS which was otherwise hard. In the prior work, the authors use asymmetric transformations which convert the problem of approximate MIPS into the problem of approximate near neighbor search which can be efficiently solved using hashing. In this work, we provide a different transformation which converts the problem of approximate MIPS into the problem of approximate cosine similarity s…
Normandie Univ（诺曼底大学）；ENSICAEN（卡昂高等工程师学院）；UNICAEN（卡昂大学）；CNRS（法国国家科学研究中心）；GREYC（格雷克实验室）；Free University of Bozen-Bolzano（博尔扎诺自由大学）；Faculty of Computer Science（计算机科学学院）
The graph edit distance (GED) measures the dissimilarity between two graphs as the minimal cost of a sequence of elementary operations transforming one graph into another. This measure is fundamental in many areas such as structural pattern recognition or classification. However, exactly computing GED is NP-hard. Among different classes of heuristic algorithms that were proposed to compute approximate solutions, local search based algorithms provide the tightest upper bounds for GED. In this paper, we present K-REFINE and RANDPOST. K-REFINE generalizes and improves an existing local search alg…
