We study approximation algorithms for variants of the \emph{median string} problem, which asks for a string that minimizes the sum of edit distances from a given set of $m$ strings of length $n$. Only the straightforward $2$-approximation is known for this NP-hard problem. This problem is motivated e.g.~by computational biology, and belongs to the class of median problems (over different metric spaces), which are fundamental tasks in data analysis.
Our main result is for the Ulam metric, where all strings are permutations over $[n]$ and each edit operation moves a symbol (deletion plus insertion). We devise for this problem an algorithms that breaks the $2$-approximation barrier, i.e., computes a $(2-δ)$-approximate median permutation for some constant $δ>0$ in time $\tilde{O}(nm^2+n^3)$. We further use these techniques to achieve a $(2-δ)$ approximation for the median string problem in the special case where the median is restricted to length $n$ and the optimal objective is large $Ω(mn)$.
We also design an approximation algorithm for the following probabilistic model of the Ulam median: the input consists of $m$ perturbations of an (unknown) permutation $x$, each generated by moving every symbol to a random position with probability (a parameter) $ε>0$. Our algorithm computes with high probability a $(1+o(1/ε))$-approximate median permutation in time $O(mn^2+n^3)$.
We present a novel approximation algorithm for $k$-median that achieves an approximation guarantee of $1+\sqrt{3}+ε$, improving upon the decade-old ratio of $3+ε$. Our approach is based on two components, each of which, we believe, is of independent interest. First, we show that in order to give an $α$-approximation algorithm for $k$-median, it is sufficient to give a \emph{pseudo-approximation algorithm} that finds an $α$-approximate solution by opening $k+O(1)$ facilities. This is a rather surprising result as there exist instances for which opening $k+1$ facilities may lead to a significant…
We describe two quantum algorithms to approximate the mean value of a black-box function. The first algorithm is novel and asymptotically optimal while the second is a variation on an earlier algorithm due to Aharonov. Both algorithms have their own strengths and caveats and may be relevant in different contexts. We then propose a new algorithm for approximating the median of a set of points over an arbitrary distance function.
In the capacitated $k$-median (\CKM) problem, we are given a set $F$ of facilities, each facility $i \in F$ with a capacity $u_i$, a set $C$ of clients, a metric $d$ over $F \cup C$ and an integer $k$. The goal is to open $k$ facilities in $F$ and connect the clients $C$ to the open facilities such that each facility $i$ is connected by at most $u_i$ clients, so as to minimize the total connection cost. In this paper, we give the first constant approximation for \CKM, that only violates the cardinality constraint by a factor of $1+ε$. This generalizes the result of [Li15], which only works for…
We investigate the fine-grained complexity of approximating the classical $k$-median / $k$-means clustering problems in general metric spaces. We show how to improve the approximation factors to $(1+2/e+\varepsilon)$ and $(1+8/e+\varepsilon)$ respectively, using algorithms that run in fixed-parameter time. Moreover, we show that we cannot do better in FPT time, modulo recent complexity-theoretic conjectures.
In 2015, Driemel, Krivošija and Sohler introduced the $(k,\ell)$-median problem for clustering polygonal curves under the Fréchet distance. Given a set of input curves, the problem asks to find $k$ median curves of at most $\ell$ vertices each that minimize the sum of Fréchet distances over all input curves to their closest median curve. A major shortcoming of their algorithm is that the input curves are restricted to lie on the real line. In this paper, we present a randomized bicriteria-approximation algorithm that works for polygonal curves in $\mathbb{R}^d$ and achieves approximation facto…
We study the power of uniform sampling for $k$-Median in various metric spaces. We relate the query complexity for approximating $k$-Median, to a key parameter of the dataset, called the balancedness $β\in (0, 1]$ (with $1$ being perfectly balanced). We show that any algorithm must make $Ω(1 / β)$ queries to the point set in order to achieve $O(1)$-approximation for $k$-Median. This particularly implies existing constructions of coresets, a popular data reduction technique, cannot be query-efficient. On the other hand, we show a simple uniform sample of $\mathrm{poly}(k ε^{-1} β^{-1})$ points …
Zwick's $(1+\varepsilon)$-approximation algorithm for the All Pairs Shortest Path (APSP) problem runs in time $\widetilde{O}(\frac{n^ω}{\varepsilon} \log{W})$, where $ω\le 2.373$ is the exponent of matrix multiplication and $W$ denotes the largest weight. This can be used to approximate several graph characteristics including the diameter, radius, median, minimum-weight triangle, and minimum-weight cycle in the same time bound. Since Zwick's algorithm uses the scaling technique, it has a factor $\log W$ in the running time. In this paper, we study whether APSP and related problems admit approx…
Ben-Gurion University（本-古里安大学）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）
We introduce the $(p,q)$-Fair Clustering problem. In this problem, we are given a set of points $P$ and a collection of different weight functions $W$. We would like to find a clustering which minimizes the $\ell_q$-norm of the vector over $W$ of the $\ell_p$-norms of the weighted distances of points in $P$ from the centers. This generalizes various clustering problems, including Socially Fair $k$-Median and $k$-Means, and is closely connected to other problems such as Densest $k$-Subgraph and Min $k$-Union. We utilize convex programming techniques to approximate the $(p,q)$-Fair Clustering pr…
This paper resolves one of the longest standing basic problems in the streaming computational model. Namely, optimal construction of quantile sketches. An $\varepsilon$ approximate quantile sketch receives a stream of items $x_1,\ldots,x_n$ and allows one to approximate the rank of any query up to additive error $\varepsilon n$ with probability at least $1-δ$. The rank of a query $x$ is the number of stream items such that $x_i \le x$. The minimal sketch size required for this task is trivially at least $1/\varepsilon$. Felber and Ostrovsky obtain a $O((1/\varepsilon)\log(1/\varepsilon))$ spac…
In this work, we study the $k$-median and $k$-means clustering problems when the data is distributed across many servers and can contain outliers. While there has been a lot of work on these problems for worst-case instances, we focus on gaining a finer understanding through the lens of beyond worst-case analysis. Our main motivation is the following: for many applications such as clustering proteins by function or clustering communities in a social network, there is some unknown target clustering, and the hope is that running a $k$-median or $k$-means algorithm will produce clusterings which …
