A simple 2(1-1/
            <i>l</i>
            ) factor distributed approximation algorithm for steiner tree in the
            <i>CONGEST</i>
            model

Saikia, Parikshit; Karmakar, Sushanta

doi:10.1145/3288599.3288629

Cited by 4 publications

(14 citation statements)

References 33 publications

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“…In the Steiner tree problem, the input is a graph G = (V, E) with weights on the edges, and a set of terminals S ⊆ V , and the goal is to find a tree of minimum cost that spans all the terminals. While several efficient approximation algorithm to the problem exist [29,36,46], we show that solving the problem exactly requires near-quadratic number of rounds. Our lower bound is obtained using a reduction from our MDS lower bound graph construction, and we first formally define the notion of reductions between families of lower bound graphs.…”

Section: Minimum Steiner Treementioning

confidence: 98%

Hardness of Distributed Optimization

Bacrach

Censor-Hillel

Dory

et al. 2019

Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing

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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...

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Section: Minimum Steiner Treementioning

confidence: 98%

Hardness of Distributed Optimization

Bacrach

Censor-Hillel

Dory

et al. 2019

Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing

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“…Specifically they showed that the lower bound round complexity for graph connectivity problem in the BCCM(1) 2 , which is Ω(log n), holds for both deterministic as well as constant-error randomized Monte Carlo algorithms [20]. Despite the fact that the ST problem has been extensively studied in the CONGEST model of distributed computing [22,23,24,25,26], to the best of our knowledge, such a study has not been carried out in the CCM. The best deterministic round complexity for solving the ST problem in the CONGEST model was recently proposed by Saikia and Karmakar [26], which is O(S + √ n log * n) with the approximation factor of 2(1 − 1/ ), where S is the shortest path diameter 3 (the definition is deferred to Section 2) of a graph and is the number of terminal leaf nodes in the optimal ST, which improves the previous best round complexity of the ST problem [25].…”

Section: Introductionmentioning

confidence: 99%

“…Despite the fact that the ST problem has been extensively studied in the CONGEST model of distributed computing [22,23,24,25,26], to the best of our knowledge, such a study has not been carried out in the CCM. The best deterministic round complexity for solving the ST problem in the CONGEST model was recently proposed by Saikia and Karmakar [26], which is O(S + √ n log * n) with the approximation factor of 2(1 − 1/ ), where S is the shortest path diameter 3 (the definition is deferred to Section 2) of a graph and is the number of terminal leaf nodes in the optimal ST, which improves the previous best round complexity of the ST problem [25]. The MST problem is a special case of the ST problem and has been extensively studied in the CONGEST model as well as in the CCM.…”

Section: Introductionmentioning

confidence: 99%

“…Given a connected undirected weighted graph G = (V, E, w) and a terminal set Z ⊆ V, there exists an algorithm that computes an ST in Õ(n 1/3 ) rounds and Õ(n 7/3 ) messages in the CCM with an approximation factor of 2(1 − 1/ ), where is the number of terminal leaf nodes in the optimal ST. The proposed STCCM-A algorithm is inspired by an algorithm proposed in [26]. It consists of four steps (each step is a small distributed algorithm)-the first step is to build a SPF G F of the input graph G for a given terminal set Z, which is essentially a partition of the graph G into disjoint trees: Each partition contains exactly one terminal and a subset of non-terminals.…”

Section: Introductionmentioning

confidence: 99%

“…Lenzen and Patt-Shamir [25] [25], in which one is deterministic and the other one is randomized reduce to Õ(S + √ min{S t, n}) and Õ(D + √ n) respectively. Saikia and Karmakar [26] proposed a deterministic distributed 2(1 − 1/ )-factor approximation algorithm for the ST problem in the CONGEST model with the round and message complexities of…”

Section: Introductionmentioning

confidence: 99%

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Distributed Approximation Algorithms for Steiner Tree in the $\mathcal{CONGESTED}$ $\mathcal{CLIQUE}$

Saikia¹,

Karmakar²

2019

Preprint

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The Steiner tree problem is one of the fundamental and classical problems in combinatorial optimization.In this paper we study this problem in the CONGEST ED CLIQUE model of distributed computing and present two deterministic distributed approximation algorithms for the same. The first algorithm computes a Steiner tree in Õ(n 1/3 ) rounds and Õ(n 7/3 ) messages for a given connected undirected weighted graph of n nodes. Note here that Õ(•) notation hides polylogarithmic factors in n. The second one computes a Steiner tree in O(S + log log n) rounds and O(S (n − t) 2 + n 2 ) messages, where S and t are the shortest path diameter and the number of terminal nodes respectively in the given input graph. Both the algorithms admit an approximation factor of 2(1 − 1/ ), where is the number of terminal leaf nodes in the optimal Steiner tree. For graphs with S = ω(n 1/3 log n), the first algorithm exhibits better performance than the second one in terms of the round complexity. On the other hand, for graphs with S = õ(n 1/3 ), the second algorithm outperforms the first one in terms of the round complexity. In fact when S = O(log log n) then the second algorithm admits a round complexity of O(log log n) and message complexity of Õ(n 2 ). To the best of our knowledge, this is the first work to study the Steiner tree problem in the CONGEST ED CLIQUE model.

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Round-Message Trade-Off in Distributed Steiner Tree Construction in the CONGEST Model

Saikia

Karmakar

2019

Distributed Computing and Internet Technology

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A simple 2(1-1/ l ) factor distributed approximation algorithm for steiner tree in the CONGEST model

Cited by 4 publications

References 33 publications

Hardness of Distributed Optimization

Hardness of Distributed Optimization

Distributed Approximation Algorithms for Steiner Tree in the $\mathcal{CONGESTED}$ $\mathcal{CLIQUE}$

Round-Message Trade-Off in Distributed Steiner Tree Construction in the CONGEST Model

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