163,310 research outputs found

    Online unit clustering in higher dimensions

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    We revisit the online Unit Clustering and Unit Covering problems in higher dimensions: Given a set of nn points in a metric space, that arrive one by one, Unit Clustering asks to partition the points into the minimum number of clusters (subsets) of diameter at most one; while Unit Covering asks to cover all points by the minimum number of balls of unit radius. In this paper, we work in Rd\mathbb{R}^d using the LL_\infty norm. We show that the competitive ratio of any online algorithm (deterministic or randomized) for Unit Clustering must depend on the dimension dd. We also give a randomized online algorithm with competitive ratio O(d2)O(d^2) for Unit Clustering}of integer points (i.e., points in Zd\mathbb{Z}^d, dNd\in \mathbb{N}, under LL_{\infty} norm). We show that the competitive ratio of any deterministic online algorithm for Unit Covering is at least 2d2^d. This ratio is the best possible, as it can be attained by a simple deterministic algorithm that assigns points to a predefined set of unit cubes. We complement these results with some additional lower bounds for related problems in higher dimensions.Comment: 15 pages, 4 figures. A preliminary version appeared in the Proceedings of the 15th Workshop on Approximation and Online Algorithms (WAOA 2017

    The Complexity of Distributed Approximation of Packing and Covering Integer Linear Programs

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    In this paper, we present a low-diameter decomposition algorithm in the LOCAL model of distributed computing that succeeds with probability 11/poly(n)1 - 1/poly(n). Specifically, we show how to compute an (ϵ,O(lognϵ))\left(\epsilon, O\left(\frac{\log n}{\epsilon}\right)\right) low-diameter decomposition in O(log3(1/ϵ)lognϵ)O\left(\frac{\log^3(1/\epsilon)\log n}{\epsilon}\right) round Further developing our techniques, we show new distributed algorithms for approximating general packing and covering integer linear programs in the LOCAL model. For packing problems, our algorithm finds an (1ϵ)(1-\epsilon)-approximate solution in O(log3(1/ϵ)lognϵ)O\left(\frac{\log^3 (1/\epsilon) \log n}{\epsilon}\right) rounds with probability 11/poly(n)1 - 1/poly(n). For covering problems, our algorithm finds an (1+ϵ)(1+\epsilon)-approximate solution in O((loglogn+log(1/ϵ))3lognϵ)O\left(\frac{\left(\log \log n + \log (1/\epsilon)\right)^3 \log n}{\epsilon}\right) rounds with probability 11/poly(n)1 - 1/poly(n). These results improve upon the previous O(log3nϵ)O\left(\frac{\log^3 n}{\epsilon}\right)-round algorithm by Ghaffari, Kuhn, and Maus [STOC 2017] which is based on network decompositions. Our algorithms are near-optimal for many fundamental combinatorial graph optimization problems in the LOCAL model, such as minimum vertex cover and minimum dominating set, as their (1±ϵ)(1\pm \epsilon)-approximate solutions require Ω(lognϵ)\Omega\left(\frac{\log n}{\epsilon}\right) rounds to compute.Comment: To appear in PODC 202

    On the Complexity of Local Distributed Graph Problems

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    This paper is centered on the complexity of graph problems in the well-studied LOCAL model of distributed computing, introduced by Linial [FOCS '87]. It is widely known that for many of the classic distributed graph problems (including maximal independent set (MIS) and (Δ+1)(\Delta+1)-vertex coloring), the randomized complexity is at most polylogarithmic in the size nn of the network, while the best deterministic complexity is typically 2O(logn)2^{O(\sqrt{\log n})}. Understanding and narrowing down this exponential gap is considered to be one of the central long-standing open questions in the area of distributed graph algorithms. We investigate the problem by introducing a complexity-theoretic framework that allows us to shed some light on the role of randomness in the LOCAL model. We define the SLOCAL model as a sequential version of the LOCAL model. Our framework allows us to prove completeness results with respect to the class of problems which can be solved efficiently in the SLOCAL model, implying that if any of the complete problems can be solved deterministically in logO(1)n\log^{O(1)} n rounds in the LOCAL model, we can deterministically solve all efficient SLOCAL-problems (including MIS and (Δ+1)(\Delta+1)-coloring) in logO(1)n\log^{O(1)} n rounds in the LOCAL model. We show that a rather rudimentary looking graph coloring problem is complete in the above sense: Color the nodes of a graph with colors red and blue such that each node of sufficiently large polylogarithmic degree has at least one neighbor of each color. The problem admits a trivial zero-round randomized solution. The result can be viewed as showing that the only obstacle to getting efficient determinstic algorithms in the LOCAL model is an efficient algorithm to approximately round fractional values into integer values

    Families of nested completely regular codes and distance-regular graphs

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    In this paper infinite families of linear binary nested completely regular codes are constructed. They have covering radius ρ\rho equal to 33 or 44, and are 1/2i1/2^i-th parts, for i{1,,u}i\in\{1,\ldots,u\} of binary (respectively, extended binary) Hamming codes of length n=2m1n=2^m-1 (respectively, 2m2^m), where m=2um=2u. In the usual way, i.e., as coset graphs, infinite families of embedded distance-regular coset graphs of diameter DD equal to 33 or 44 are constructed. In some cases, the constructed codes are also completely transitive codes and the corresponding coset graphs are distance-transitive

    Highly saturated packings and reduced coverings

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    We introduce and study certain notions which might serve as substitutes for maximum density packings and minimum density coverings. A body is a compact connected set which is the closure of its interior. A packing P\cal P with congruent replicas of a body KK is nn-saturated if no n1n-1 members of it can be replaced with nn replicas of KK, and it is completely saturated if it is nn-saturated for each n1n\ge 1. Similarly, a covering C\cal C with congruent replicas of a body KK is nn-reduced if no nn members of it can be replaced by n1n-1 replicas of KK without uncovering a portion of the space, and it is completely reduced if it is nn-reduced for each n1n\ge 1. We prove that every body KK in dd-dimensional Euclidean or hyperbolic space admits both an nn-saturated packing and an nn-reduced covering with replicas of KK. Under some assumptions on KEdK\subset \mathbb{E}^d (somewhat weaker than convexity), we prove the existence of completely saturated packings and completely reduced coverings, but in general, the problem of existence of completely saturated packings and completely reduced coverings remains unsolved. Also, we investigate some problems related to the the densities of nn-saturated packings and nn-reduced coverings. Among other things, we prove that there exists an upper bound for the density of a d+2d+2-reduced covering of Ed\mathbb{E}^d with congruent balls, and we produce some density bounds for the nn-saturated packings and nn-reduced coverings of the plane with congruent circles

    Estimation of instrinsic dimension via clustering

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    The problem of estimating the intrinsic dimension of a set of points in high dimensional space is a critical issue for a wide range of disciplines, including genomics, finance, and networking. Current estimation techniques are dependent on either the ambient or intrinsic dimension in terms of computational complexity, which may cause these methods to become intractable for large data sets. In this paper, we present a clustering-based methodology that exploits the inherent self-similarity of data to efficiently estimate the intrinsic dimension of a set of points. When the data satisfies a specified general clustering condition, we prove that the estimated dimension approaches the true Hausdorff dimension. Experiments show that the clustering-based approach allows for more efficient and accurate intrinsic dimension estimation compared with all prior techniques, even when the data does not conform to obvious self-similarity structure. Finally, we present empirical results which show the clustering-based estimation allows for a natural partitioning of the data points that lie on separate manifolds of varying intrinsic dimension

    GraphCombEx: A Software Tool for Exploration of Combinatorial Optimisation Properties of Large Graphs

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    We present a prototype of a software tool for exploration of multiple combinatorial optimisation problems in large real-world and synthetic complex networks. Our tool, called GraphCombEx (an acronym of Graph Combinatorial Explorer), provides a unified framework for scalable computation and presentation of high-quality suboptimal solutions and bounds for a number of widely studied combinatorial optimisation problems. Efficient representation and applicability to large-scale graphs and complex networks are particularly considered in its design. The problems currently supported include maximum clique, graph colouring, maximum independent set, minimum vertex clique covering, minimum dominating set, as well as the longest simple cycle problem. Suboptimal solutions and intervals for optimal objective values are estimated using scalable heuristics. The tool is designed with extensibility in mind, with the view of further problems and both new fast and high-performance heuristics to be added in the future. GraphCombEx has already been successfully used as a support tool in a number of recent research studies using combinatorial optimisation to analyse complex networks, indicating its promise as a research software tool
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