Algorithms and Hardness for Diameter in Dynamic Graphs

The diameter, radius and eccentricities are natural graph parameters. While these problems have been studied extensively, there are no known dynamic algorithms for them beyond the ones that follow from trivial recomputation after each update or from solving dynamic All-Pairs Shortest Paths (APSP), which is very computationally intensive. This is the situation for dynamic approximation algorithms as well, and even if only edge insertions or edge deletions need to be supported. This paper provides a comprehensive study of the dynamic approximation of Diameter, Radius and Eccentricities, providing both conditional lower bounds, and new algorithms whose bounds are optimal under popular hypotheses in fine-grained complexity. Some of the highlights include: - Under popular hardness hypotheses, there can be no significantly better fully dynamic approximation algorithms than recomputing the answer after each update, or maintaining full APSP. - Nearly optimal partially dynamic (incremental/decremental) algorithms can be achieved via efficient reductions to (incremental/decremental) maintenance of Single-Source Shortest Paths. For instance, a nearly (3/2+epsilon)-approximation to Diameter in directed or undirected n-vertex, m-edge graphs can be maintained decrementally in total time m^{1+o(1)}sqrt{n}/epsilon^2. This nearly matches the static 3/2-approximation algorithm for the problem that is known to be conditionally optimal

Dynamic algorithms for k-center on graphs

In this paper we give the first efficient algorithms for the $k$-center problem on dynamic graphs undergoing edge updates. In this problem, the goal is to partition the input into $k$ sets by choosing $k$ centers such that the maximum distance from any data point to the closest center is minimized. It is known that it is NP-hard to get a better than $2$ approximation for this problem. While in many applications the input may naturally be modeled as a graph, all prior works on $k$-center problem in dynamic settings are on metrics. In this paper, we give a deterministic decremental $(2+\epsilon)$-approximation algorithm and a randomized incremental $(4+\epsilon)$-approximation algorithm, both with amortized update time $kn^{o(1)}$ for weighted graphs. Moreover, we show a reduction that leads to a fully dynamic $(2+\epsilon)$-approximation algorithm for the $k$-center problem, with worst-case update time that is within a factor $k$ of the state-of-the-art upper bound for maintaining $(1+\epsilon)$-approximate single-source distances in graphs. Matching this bound is a natural goalpost because the approximate distances of each vertex to its center can be used to maintain a $(2+\epsilon)$-approximation of the graph diameter and the fastest known algorithms for such a diameter approximation also rely on maintaining approximate single-source distances

Fast Distributed Approximation for TAP and 2-Edge-Connectivity

The tree augmentation problem (TAP) is a fundamental network design problem, in which the input is a graph $G$ and a spanning tree $T$ for it, and the goal is to augment $T$ with a minimum set of edges $Aug$ from $G$, such that $T \cup Aug$ is 2-edge-connected. TAP has been widely studied in the sequential setting. The best known approximation ratio of 2 for the weighted case dates back to the work of Frederickson and J\'{a}J\'{a}, SICOMP 1981. Recently, a 3/2-approximation was given for unweighted TAP by Kortsarz and Nutov, TALG 2016. Recent breakthroughs give an approximation of 1.458 for unweighted TAP [Grandoni et al., STOC 2018], and approximations better than 2 for bounded weights [Adjiashvili, SODA 2017; Fiorini et al., SODA 2018]. In this paper, we provide the first fast distributed approximations for TAP. We present a distributed $2$-approximation for weighted TAP which completes in $O(h)$ rounds, where $h$ is the height of $T$. When $h$ is large, we show a much faster 4-approximation algorithm for the unweighted case, completing in $O(D+\sqrt{n}\log^*{n})$ rounds, where $n$ is the number of vertices and $D$ is the diameter of $G$. Immediate consequences of our results are an $O(D)$-round 2-approximation algorithm for the minimum size 2-edge-connected spanning subgraph, which significantly improves upon the running time of previous approximation algorithms, and an $O(h_{MST}+\sqrt{n}\log^{*}{n})$-round 3-approximation algorithm for the weighted case, where $h_{MST}$ is the height of the MST of the graph. Additional applications are algorithms for verifying 2-edge-connectivity and for augmenting the connectivity of any connected spanning subgraph to 2. Finally, we complement our study with proving lower bounds for distributed approximations of TAP

Parameterized Distributed Algorithms

In this work, we initiate a thorough study of graph optimization problems parameterized by the output size in the distributed setting. In such a problem, an algorithm decides whether a solution of size bounded by k exists and if so, it finds one. We study fundamental problems, including Minimum Vertex Cover (MVC), Maximum Independent Set (MaxIS), Maximum Matching (MaxM), and many others, in both the LOCAL and CONGEST distributed computation models. We present lower bounds for the round complexity of solving parameterized problems in both models, together with optimal and near-optimal upper bounds. Our results extend beyond the scope of parameterized problems. We show that any LOCAL (1+epsilon)-approximation algorithm for the above problems must take Omega(epsilon^{-1}) rounds. Joined with the (epsilon^{-1}log n)^{O(1)} rounds algorithm of [Ghaffari et al., 2017] and the Omega (sqrt{(log n)/(log log n)}) lower bound of [Fabian Kuhn et al., 2016], the lower bounds match the upper bound up to polynomial factors in both parameters. We also show that our parameterized approach reduces the runtime of exact and approximate CONGEST algorithms for MVC and MaxM if the optimal solution is small, without knowing its size beforehand. Finally, we propose the first o(n^2) rounds CONGEST algorithms that approximate MVC within a factor strictly smaller than 2

A Practical Parallel Algorithm for Diameter Approximation of Massive Weighted Graphs

We present a space and time efficient practical parallel algorithm for approximating the diameter of massive weighted undirected graphs on distributed platforms supporting a MapReduce-like abstraction. The core of the algorithm is a weighted graph decomposition strategy generating disjoint clusters of bounded weighted radius. Theoretically, our algorithm uses linear space and yields a polylogarithmic approximation guarantee; moreover, for important practical classes of graphs, it runs in a number of rounds asymptotically smaller than those required by the natural approximation provided by the state-of-the-art $\Delta$-stepping SSSP algorithm, which is its only practical linear-space competitor in the aforementioned computational scenario. We complement our theoretical findings with an extensive experimental analysis on large benchmark graphs, which demonstrates that our algorithm attains substantial improvements on a number of key performance indicators with respect to the aforementioned competitor, while featuring a similar approximation ratio (a small constant less than 1.4, as opposed to the polylogarithmic theoretical bound)

Fast approximation of centrality and distances in hyperbolic graphs

We show that the eccentricities (and thus the centrality indices) of all vertices of a $\delta$-hyperbolic graph $G=(V,E)$ can be computed in linear time with an additive one-sided error of at most $c\delta$, i.e., after a linear time preprocessing, for every vertex $v$ of $G$ one can compute in $O(1)$ time an estimate $\hat{e}(v)$ of its eccentricity $ecc_G(v)$ such that $ecc_G(v)\leq \hat{e}(v)\leq ecc_G(v)+ c\delta$ for a small constant $c$. We prove that every $\delta$-hyperbolic graph $G$ has a shortest path tree, constructible in linear time, such that for every vertex $v$ of $G$, $ecc_G(v)\leq ecc_T(v)\leq ecc_G(v)+ c\delta$. These results are based on an interesting monotonicity property of the eccentricity function of hyperbolic graphs: the closer a vertex is to the center of $G$, the smaller its eccentricity is. We also show that the distance matrix of $G$ with an additive one-sided error of at most $c'\delta$ can be computed in $O(|V|^2\log^2|V|)$ time, where $c'< c$ is a small constant. Recent empirical studies show that many real-world graphs (including Internet application networks, web networks, collaboration networks, social networks, biological networks, and others) have small hyperbolicity. So, we analyze the performance of our algorithms for approximating centrality and distance matrix on a number of real-world networks. Our experimental results show that the obtained estimates are even better than the theoretical bounds.Comment: arXiv admin note: text overlap with arXiv:1506.01799 by other author