Tight Approximation Algorithms for Bichromatic Graph Diameter and Related Problems

Some of the most fundamental and well-studied graph parameters are the Diameter (the largest shortest paths distance) and Radius (the smallest distance for which a "center" node can reach all other nodes). The natural and important ST-variant considers two subsets S and T of the vertex set and lets the ST-diameter be the maximum distance between a node in S and a node in T, and the ST-radius be the minimum distance for a node of S to reach all nodes of T. The bichromatic variant is the special case in which S and T partition the vertex set. In this paper we present a comprehensive study of the approximability of ST and Bichromatic Diameter, Radius, and Eccentricities, and variants, in graphs with and without directions and weights. We give the first nontrivial approximation algorithms for most of these problems, including time/accuracy trade-off upper and lower bounds. We show that nearly all of our obtained bounds are tight under the Strong Exponential Time Hypothesis (SETH), or the related Hitting Set Hypothesis. For instance, for Bichromatic Diameter in undirected weighted graphs with m edges, we present an O~(m^{3/2}) time 5/3-approximation algorithm, and show that under SETH, neither the running time, nor the approximation factor can be significantly improved while keeping the other unchanged

Distance Computations in the Hybrid Network Model via Oracle Simulations

The Hybrid network model was introduced in [Augustine et al., SODA '20] for laying down a theoretical foundation for networks which combine two possible modes of communication: One mode allows high-bandwidth communication with neighboring nodes, and the other allows low-bandwidth communication over few long-range connections at a time. This fundamentally abstracts networks such as hybrid data centers, and class-based software-defined networks. Our technical contribution is a \emph{density-aware} approach that allows us to simulate a set of \emph{oracles} for an overlay skeleton graph over a Hybrid network. As applications of our oracle simulations, with additional machinery that we provide, we derive fast algorithms for fundamental distance-related tasks. One of our core contributions is an algorithm in the Hybrid model for computing \emph{exact} weighted shortest paths from $\tilde O(n^{1/3})$ sources which completes in $\tilde O(n^{1/3})$ rounds w.h.p. This improves, in both the runtime and the number of sources, upon the algorithm of [Kuhn and Schneider, PODC '20], which computes shortest paths from a single source in $\tilde O(n^{2/5})$ rounds w.h.p. We additionally show a 2-approximation for weighted diameter and a $(1+\epsilon)$-approximation for unweighted diameter, both in $\tilde O(n^{1/3})$ rounds w.h.p., which is comparable to the $\tilde \Omega(n^{1/3})$ lower bound of [Kuhn and Schneider, PODC '20] for a $(2-\epsilon)$-approximation for weighted diameter and an exact unweighted diameter. We also provide fast distance \emph{approximations} from multiple sources and fast approximations for eccentricities.Comment: To appear in STACS 202

Deterministic Distributed Algorithms and Lower Bounds in the Hybrid Model

The HYBRID model was recently introduced by Augustine et al. [John Augustine et al., 2020] in order to characterize from an algorithmic standpoint the capabilities of networks which combine multiple communication modes. Concretely, it is assumed that the standard LOCAL model of distributed computing is enhanced with the feature of all-to-all communication, but with very limited bandwidth, captured by the node-capacitated clique (NCC). In this work we provide several new insights on the power of hybrid networks for fundamental problems in distributed algorithms. First, we present a deterministic algorithm which solves any problem on a sparse n-node graph in ??(?n) rounds of HYBRID, where the notation ??(?) suppresses polylogarithmic factors of n. We combine this primitive with several sparsification techniques to obtain efficient distributed algorithms for general graphs. Most notably, for the all-pairs shortest paths problem we give deterministic (1 + ?)- and log n/log log n-approximate algorithms for unweighted and weighted graphs respectively with round complexity ??(?n) in HYBRID, closely matching the performance of the state of the art randomized algorithm of Kuhn and Schneider [Kuhn and Schneider, 2020]. Moreover, we make a connection with the Ghaffari-Haeupler framework of low-congestion shortcuts [Mohsen Ghaffari and Bernhard Haeupler, 2016], leading - among others - to a (1 + ?)-approximate algorithm for Min-Cut after ?(polylog (n)) rounds, with high probability, even if we restrict local edges to transfer ?(log n) bits per round. Finally, we prove via a reduction from the set disjointness problem that ??(n^{1/3}) rounds are required to determine the radius of an unweighted graph, as well as a (3/2 - ?)-approximation for weighted graphs. As a byproduct, we show an ??(n) round-complexity lower bound for computing a (4/3 - ?)-approximation of the radius in the broadcast variant of the congested clique, even for unweighted graphs