Everywhere-Sparse Spanners via Dense Subgraphs

E, Eden Chlamt #x; Dinitz, Michael; Krauthgamer, Robert

doi:10.1109/focs.2012.61

Cited by 36 publications

(58 citation statements)

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“…As mentioned, we look at random distinguishing problems of the form studied by Bhaskara et al [4] and Chlamtáč et al [8]. Define the log-density of a graph on n nodes to be log n (D avg ), where D avg is the average degree.…”

Section: Our Results and Techniquesmentioning

confidence: 99%

“…One of the problems considered in [4,8] is the Dense vs Random problem, which is parameterized by k and constants 0 < α, β < 1: Given a graph G, distinguish between the following two cases: 1) G = G(n, p) where p = n α−1 (and thus the graph has log-density concentrated around α), and 2) G is adversarially chosen so that the densest k-subgraph has logdensity β where k β ≫ pk (and thus the average degree inside this subgraph is approximately k β ). The following conjecture was explicitly given in [8], and implies that the known algorithms for DkS and SmES are tight: This conjecture can quite naturally be extended to hypergraphs. Let G n,p,r denote the distribution over r-uniform hypergraphs obtained by choosing every subset of cardinality r to be a hyperedge independently with probability p. Define the Hypergraph Dense vs Random problem as follows, again parameterized by k and constants 0 < α, β < r−1.…”

Section: Our Results and Techniquesmentioning

confidence: 99%

“…While our aim is to apply the framework of [4] to SSBVE, we note that SSBVE and DkS (or the minimization version SmES) differ from each other in important ways, making it impossible to straightforwardly apply the ideas from [4] or [8]. The asymmetry between U and V in SSBVE requires us to fundamentally change our approach; loosely speaking, in SSBVE, we are looking not for an arbitrary dense subgraph of a bipartite graph, but rather for a dense subgraph of the form S ∪ N (S) (where S is a subset of U of size k), since once we choose the set S ⊂ U we must take into account all neighbors of S in V .…”

Section: Our Results and Techniquesmentioning

confidence: 99%

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Minimizing the Union: Tight Approximations for Small Set Bipartite Vertex Expansion

Chlamtáč

Dinitz

Makarychev³

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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In the Minimum k-Union problem (MkU) we are given a set system with n sets and are asked to select k sets in order to minimize the size of their union. Despite being a very natural problem, it has received surprisingly little attention: the only known approximation algorithm is an O( √ n)-approximation due to [Chlamtáč et al APPROX '16]. This problem can also be viewed as the bipartite version of the Small Set Vertex Expansion problem (SSVE), which we call the Small Set Bipartite Vertex Expansion problem (SSBVE). SSVE, in which we are asked to find a set of k nodes to minimize their vertex expansion, has not been as well studied as its edge-based counterpart Small Set Expansion (SSE), but has recently received significant attention, e.g. [Louis-Makarychev APPROX '15]. However, due to the connection to Unique Games and hardness of approximation the focus has mostly been on sets of size k = Ω(n), while we focus on the case of general k, for which no polylogarithmic approximation is known.We improve the upper bound for this problem by giving an n 1/4+ε approximation for SSBVE for any constant ε > 0. Our algorithm follows in the footsteps of Densest k-Subgraph (DkS) and related problems, by designing a tight algorithm for random models, and then extending it to give the same guarantee for arbitrary instances. Moreover, we show that this is tight under plausible complexity conjectures: it cannot be approximated better than O(n 1/4 ) assuming an extension of the so-called "Dense versus Random" conjecture for DkS to hypergraphs.In addition to conjectured hardness via our reduction, we show that the same lower bound is also matched by an integrality gap for a super-constant number of rounds of the Sherali-Adams LP hierarchy, and an even worse integrality gap for the natural SDP relaxation. Finally, we note that there exists a simple bicriteriã O( √ n) approximation for the more general SSVE prob- * Ben Gurion University.

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Section: Our Results and Techniquesmentioning

confidence: 99%

Section: Our Results and Techniquesmentioning

confidence: 99%

Section: Our Results and Techniquesmentioning

confidence: 99%

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Minimizing the Union: Tight Approximations for Small Set Bipartite Vertex Expansion

Chlamtáč

Dinitz

Makarychev³

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

Self Cite

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“…These approximation ratios are matched by a recent distributed O(k log n)-round algorithm, that uses only polynomial local computations [22]. Approximation algorithms are given also for pairwise spanners and distance preservers [14], for spanners with lowest maximum degree [13,15,22,47], for fault-tolerant spanners [21,23], and more.…”

Section: Additional Related Workmentioning

confidence: 99%

Distributed Spanner Approximation

Censor-Hillel

Dory

2018

Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing

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We address the fundamental network design problem of constructing approximate minimum spanners. Our contributions are for the distributed setting, providing both algorithmic and hardness results.Our main hardness result shows that an α-approximation for the minimum directed kspanner problem for k ≥ 5 requires Ω(n/ √ α log n) rounds using deterministic algorithms or Ω( √ n/ √ α log n) rounds using randomized ones, in the Congest model of distributed computing. Combined with the constant-round O(n )-approximation algorithm in the Local model of [Barenboim, Elkin and Gavoille, 2016], as well as a polylog-round (1 + )-approximation algorithm in the Local model that we show here, our lower bounds for the Congest model imply a strict separation between the Local and Congest models. Notably, to the best of our knowledge, this is the first separation between these models for a local approximation problem.Similarly, a separation between the directed and undirected cases is implied. We also prove that the minimum weighted k-spanner problem for k ≥ 4 requires a near-linear number of rounds in the Congest model, for directed or undirected graphs. In addition, we show lower bounds for the minimum weighted 2-spanner problem in the Congest and Local models.On the algorithmic side, apart from the aforementioned (1 + )-approximation algorithm for minimum k-spanners, our main contribution is a new distributed construction of minimum 2-spanners that uses only polynomial local computations. Our algorithm has a guaranteed approximation ratio of O(log(m/n)) for a graph with n vertices and m edges, which matches the best known ratio for polynomial time sequential algorithms [Kortsarz and Peleg, 1994], and is tight if we restrict ourselves to polynomial local computations. An algorithm with this approximation factor was not previously known for the distributed setting. The number of rounds required for our algorithm is O(log n log ∆) w.h.p, where ∆ is the maximum degree in the graph. Our approach allows us to extend our algorithm to work also for the directed, weighted, and client-server variants of the problem. It also provides a Congest algorithm for the minimum dominating set problem, with a guaranteed O(log ∆) approximation ratio.

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“…Some of our algorithms use a method first introduced in [BCC + 10] to study Densest k-Subgraph, and which has since been used successfully for the related problems of Smallest k-Edge Subgraph [CDK12] and Small Set Bipartite Vertex Expansion [CDM17]. The method consists of the following steps, which follow in the rest of the section.…”

Section: The Log Density Methodsmentioning

confidence: 99%

Approximation Algorithms for Label Cover and The Log-Density Threshold

Chlamtáč¹,

Manurangsi²,

Moshkovitz³

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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Many known optimal NP-hardness of approximation results are reductions from a problem called LabelCover. The input is a bipartite graph G = (L, R, E) and each edge e = (x, y) ∈ E carries a projection π e that maps labels to x to labels to y. The objective is to find a labeling of the vertices that satisfies as many of the projections as possible. It is believed that the best approximation ratio efficiently achievable for Label-Cover is of the form N −c where N = nk, n is the number of vertices, k is the number of labels, and 0 < c < 1 is some constant.Inspired by a framework originally developed for Densest k-Subgraph, we propose a "log density threshold" for the approximability of Label-Cover. Specifically, we suggest the possibility that the Label-Cover approximation problem undergoes a computational phase transition at the same threshold at which local algorithms for its random counterpart fail. This threshold is N 3−2 √ 2 ≈ N −0.17 . We then design, for any ε > 0, a polynomial-time approximation algorithm for semirandom Label-Cover whose approximation ratio is N 3−2 √ 2+ε . In our semi-random model, the input graph is random (or even just expanding), and the projections on the edges are arbitrary. time algorithm whose approximation ratio is roughly N −0.233 . The previous best efficient approximation ratio was N −0.25 . We present some evidence towards an N −c threshold by constructing integrality gaps for N Ω(1) rounds of the Sum-of-squares/Lasserre hierarchy of the natural relaxation of Label Cover. For general 2CSP the "log density threshold" is N −0.25 , and we give a polynomial-time algorithm in the semi-random model whose approximation ratio is N −0.25+ε for any ε > 0.

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Everywhere-Sparse Spanners via Dense Subgraphs

Cited by 36 publications

References 47 publications

Minimizing the Union: Tight Approximations for Small Set Bipartite Vertex Expansion

Minimizing the Union: Tight Approximations for Small Set Bipartite Vertex Expansion

Distributed Spanner Approximation

Approximation Algorithms for Label Cover and The Log-Density Threshold

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