Super-Fast Distributed Algorithms for Metric Facility Location

This paper presents a distributed O(1)-approximation algorithm, with expected-$O(\log \log n)$ running time, in the $\mathcal{CONGEST}$ model for the metric facility location problem on a size-$n$ clique network. Though metric facility location has been considered by a number of researchers in low-diameter settings, this is the first sub-logarithmic-round algorithm for the problem that yields an O(1)-approximation in the setting of non-uniform facility opening costs. In order to obtain this result, our paper makes three main technical contributions. First, we show a new lower bound for metric facility location, extending the lower bound of B\u{a}doiu et al. (ICALP 2005) that applies only to the special case of uniform facility opening costs. Next, we demonstrate a reduction of the distributed metric facility location problem to the problem of computing an O(1)-ruling set of an appropriate spanning subgraph. Finally, we present a sub-logarithmic-round (in expectation) algorithm for computing a 2-ruling set in a spanning subgraph of a clique. Our algorithm accomplishes this by using a combination of randomized and deterministic sparsification.Comment: 15 pages, 2 figures. This is the full version of a paper that appeared in ICALP 201

Sensitivity Analysis of the Maximum Matching Problem

We consider the sensitivity of algorithms for the maximum matching problem against edge and vertex modifications. Algorithms with low sensitivity are desirable because they are robust to edge failure or attack. In this work, we show a randomized $(1-\epsilon)$-approximation algorithm with worst-case sensitivity $O_{\epsilon}(1)$, which substantially improves upon the $(1-\epsilon)$-approximation algorithm of Varma and Yoshida (arXiv 2020) that obtains average sensitivity $n^{O(1/(1+\epsilon^2))}$ sensitivity algorithm, and show a deterministic $1/2$-approximation algorithm with sensitivity $\exp(O(\log^*n))$ for bounded-degree graphs. We show that any deterministic constant-factor approximation algorithm must have sensitivity $\Omega(\log^* n)$. Our results imply that randomized algorithms are strictly more powerful than deterministic ones in that the former can achieve sensitivity independent of $n$ whereas the latter cannot. We also show analogous results for vertex sensitivity, where we remove a vertex instead of an edge. As an application of our results, we give an algorithm for the online maximum matching with $O_{\epsilon}(n)$ total replacements in the vertex-arrival model. By comparison, Bernstein et al. (J. ACM 2019) gave an online algorithm that always outputs the maximum matching, but only for bipartite graphs and with $O(n\log n)$ total replacements. Finally, we introduce the notion of normalized weighted sensitivity, a natural generalization of sensitivity that accounts for the weights of deleted edges. We show that if all edges in a graph have polynomially bounded weight, then given a trade-off parameter $\alpha>2$, there exists an algorithm that outputs a $\frac{1}{4\alpha}$-approximation to the maximum weighted matching in $O(m\log_{\alpha} n)$ time, with normalized weighted sensitivity $O(1)$. See paper for full abstract

New Approximability Results for the Robust k-Median Problem

We consider a robust variant of the classical $k$-median problem, introduced by Anthony et al. \cite{AnthonyGGN10}. In the \emph{Robust $k$-Median problem}, we are given an $n$-vertex metric space $(V,d)$ and $m$ client sets $\set{S_i \subseteq V}_{i=1}^m$. The objective is to open a set $F \subseteq V$ of $k$ facilities such that the worst case connection cost over all client sets is minimized; in other words, minimize $\max_{i} \sum_{v \in S_i} d(F,v)$. Anthony et al.\ showed an $O(\log m)$ approximation algorithm for any metric and APX-hardness even in the case of uniform metric. In this paper, we show that their algorithm is nearly tight by providing $\Omega(\log m/ \log \log m)$ approximation hardness, unless ${\sf NP} \subseteq \bigcap_{\delta >0} {\sf DTIME}(2^{n^{\delta}})$. This hardness result holds even for uniform and line metrics. To our knowledge, this is one of the rare cases in which a problem on a line metric is hard to approximate to within logarithmic factor. We complement the hardness result by an experimental evaluation of different heuristics that shows that very simple heuristics achieve good approximations for realistic classes of instances.Comment: 19 page

An optimal maximal independent setalgorithm for bounded-independence graphs

We present a novel distributed algorithm for the maximal independent set problem (This is an extended journal version of Schneider and Wattenhofer in Twenty-seventh annual ACM SIGACT-SIGOPS symposium on principles of distributed computing, 2008). On bounded-independence graphs our deterministic algorithm finishes in O(log* n) time, n being the number of nodes. In light of Linial's Ω(log* n) lower bound our algorithm is asymptotically optimal. Furthermore, it solves the connected dominating set problem for unit disk graphs in O(log* n) time, exponentially faster than the state-of-the-art algorithm. With a new extension our algorithm also computes a δ+1 coloring and a maximal matching in O(log* n) time, where δ is the maximum degree of the grap

Dynamic Distribution-Sensitive Point Location

We propose a dynamic data structure for the distribution-sensitive point location problem. Suppose that there is a fixed query distribution in $\mathbb{R}^2$, and we are given an oracle that can return in $O(1)$ time the probability of a query point falling into a polygonal region of constant complexity. We can maintain a convex subdivision $\cal S$ with $n$ vertices such that each query is answered in $O(\mathrm{OPT})$ expected time, where OPT is the minimum expected time of the best linear decision tree for point location in $\cal S$. The space and construction time are $O(n\log^2 n)$. An update of $\cal S$ as a mixed sequence of $k$ edge insertions and deletions takes $O(k\log^5 n)$ amortized time. As a corollary, the randomized incremental construction of the Voronoi diagram of $n$ sites can be performed in $O(n\log^5 n)$ expected time so that, during the incremental construction, a nearest neighbor query at any time can be answered optimally with respect to the intermediate Voronoi diagram at that time.Comment: To appear in Proceedings of the International Symposium of Computational Geometry, 202

On facility location problem in the local differential privacy model

In this paper we study the uncapacitated facility location problem in the model of differential privacy (DP) with uniform facility cost. Specifically, we first show that, under the hierarchically well-separated tree (HST) metrics and the super-set output setting that was introduced in [8], there is an  ∊-DP algorithm that achieves an O (¹/∊) expected multiplicative) approximation ratio; this implies an O( ^log n/_∊) approximation ratio for the general metric case, where n is the size of the input metric. These bounds improve the best-known results given by [8]. In particular, our approximation ratio for HST-metrics is independent of n, and the ratio for general metrics is independent of the aspect ratio of the input metric. On the negative side, we show that the approximation ratio of any  ∊-DP algorithm is lower bounded by Ω (1/√∊), even for instances on HST metrics with uniform facility cost, under the super-set output setting. The lower bound shows that the dependence of the approximation ratio for HST metrics on ∊ can not be removed or greatly improved. Our novel methods and techniques for both the upper and lower bound may find additional applications.CNS-2040249 - National Science Foundationhttps://proceedings.mlr.press/v151/cohen-addad22a/cohen-addad22a.pdfFirst author draf