Distributed Approximation Algorithms for Finding 2-Edge-Connected Subgraphs

2019

Distrib. Comput.

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 ∪ 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 JáJá, 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 + √ 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 M ST + √ n log * n)-round 3approximation algorithm for the weighted case, where h M ST 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. * A preliminary version of this paper appeared in OPODIS 2017.

Section: Applicationsmentioning

confidence: 63%

Section: Related Work In the Distributed Settingmentioning

confidence: 99%

“…Proof. We follow the same approach of [24]. We start by constructing an MST, which takes O(D + √ n log * n) rounds [25], and then we augment it using A wT AP in O(h M ST ) rounds.…”

Section: Applicationsmentioning

confidence: 99%

Section: Applicationsmentioning

confidence: 99%

“…Algorithms and lower bounds for weighted 2-ECSS Reference Variant Approximation Time complexity Prior work [24] 3…”

Section: Follow-up Workmentioning

confidence: 99%

See 3 more Smart Citations

Fast distributed approximation for TAP and 2-edge-connectivity

Censor-Hillel

2019

Distrib. Comput.

Distributed Approximation of Minimum k-edge-connected Spanning Subgraphs

Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing

2018

In the minimum k-edge-connected spanning subgraph (k-ECSS) problem the goal is to find the minimum weight subgraph resistant to up to k − 1 edge failures. This is a central problem in network design, and a natural generalization of the minimum spanning tree (MST) problem. While the MST problem has been studied extensively by the distributed computing community, for k ≥ 2 less is known in the distributed setting.In this paper, we present fast randomized distributed approximation algorithms for k-ECSS in the Congest model. Our first contribution is an O(D + √ n)-round O(log n)-approximation for 2-ECSS, for a graph with n vertices and diameter D. The time complexity of our algorithm is almost tight and almost matches the time complexity of the MST problem. For larger constant values of k we give an O(n)-round O(log n)-approximation. Additionally, in the special case of unweighted 3-ECSS we show how to improve the time complexity to O(D log 3 n) rounds. All our results significantly improve the time complexity of previous algorithms.1 An undirected graph G is k-edge-connected if it remains connected after the removal of any k − 1 edges. 2 All previous results use deterministic algorithms. 1 Our contributionsOur first contribution is the first sublinear algorithm for weighted 2-ECSS. Theorem 1.1. There is a distributed algorithm for weighted 2-ECSS in the Congest model that guarantees an approximation ratio of O(log n), and takes O((D + √ n) log 2 n) rounds w.h.p. The time complexity of our algorithm improves upon the previos O(h M ST + √ n log * n)-round algorithm [1], it almost matches the time complexity of the MST problem, and it is almost tight. Computing an α-approximation for weighted 2-ECSS requires Ω(D+ √ n log n ) rounds, for any polynomial function α [1]. 3 We next consider the case k > 2, and show the following. Theorem 1.2. There is a distributed algorithm for weighted k-ECSS in the Congest model with an expected approximation ratio of O(k log n), and time complexity of O(k(D log 3 n + n)) rounds. This gives the first nearly-linear time algorithm for any constant k, and improves upon the previous O(knD)-round algorithm [35]. We also show that in the special case of unweighted 3-ECSS we can improve the time complexity of the algorithm to O(D log 3 n) rounds, which improves upon the previous O(D + √ n log * n)-round algorithm [36]. Theorem 1.3. There is a distributed algorithm for unweighted 3-ECSS in the Congest model with an expected approximation ratio of O(log n), and time complexity of O(D log 3 n) rounds.Covering problems: We show that the minimum k-ECSS problem is closely related to the set cover problem. Some elements in our algorithms and analysis are inspired by the parallel set cover algorithm of Rajagopalan and Vazirani [33], and the minimum dominating set (MDS) algorithm of Jia et al. [17]. It is worth noting that our 2-ECSS algorithm guarantees the approximation ratio, where other distributed O(log n)-approximations for set cover obtain approximations that hold in expectation or w.h.p [17,24]....

Hardness of Distributed Optimization

Bacrach

Censor-Hillel

Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing

et al. 2019

This paper studies lower bounds for fundamental optimization problems in the congest model. We show that solving problems exactly in this model can be a hard task, by providing Ω(n 2 ) lower bounds for cornerstone problems, such as minimum dominating set (MDS), Hamiltonian path, Steiner tree and max-cut. These are almost tight, since all of these problems can be solved optimally in O(n 2 ) rounds. Moreover, we show that even in bounded-degree graphs and even in simple graphs with maximum degree 5 and logarithmic diameter, it holds that various tasks, such as finding a maximum independent set (MaxIS) or a minimum vertex cover, are still difficult, requiring a near-tight number ofΩ(n) rounds.Furthermore, we show that in some cases even approximations are difficult, by providing añ Ω(n 2 ) lower bound for a (7/8 + )-approximation for MaxIS, and a nearly-linear lower bound for an O(log n)-approximation for the k-MDS problem for any constant k ≥ 2, as well as for several variants of the Steiner tree problem.Our lower bounds are based on a rich variety of constructions that leverage novel observations, and reductions among problems that are specialized for the congest model. However, for several additional approximation problems, as well as for exact computation of some central problems in P , such as maximum matching and max flow, we show that such constructions cannot be designed, by which we exemplify some limitations of this framework.Lower bounds for exact computation. We show that in many cases, solving problems exactly in the congest model is hard, by providing many newΩ(n 2 ) lower bounds for fundamental optimization problems, such as MDS, max-cut, Hamiltonian path, Steiner tree and minimum 2edge-connected spanning subgraph (2-ECSS). Such results were previously known only for the minimum vertex cover (MVC), MaxIS and minimum chromatic number problems [10]. Our results are inspired by [10], but combine many new technical ingredients. In particular, one of the key components in our lower bounds are reductions between problems. After having a lower bound for MDS, a cleverly designed reduction allows us to build a new lower bound construction for Hamiltonian path. These constructions serve as a basis for our constructions for the Steiner tree and minimum 2-ECSS. We emphasize that we cannot use directly known reductions from the sequential setting, but rather we must create reductions that can be applied efficiently on lower bound constructions.To demonstrate the challenge, we now give more details about the general framework. We use the well-known framework of reductions from 2-party communication complexity, as originated in [44] and used in many additional works, e.g., [1, 9,14,17,18,47]. In communication complexity, two players, Alice and Bob, receive private input strings and their goal is to solve some problem related to their inputs, for example, decide whether their inputs are disjoint, by communicating the minimum number of bits possible. To show a lower bound for the congest model, the highlevel idea is t...