On the hardness of approximating spanners

Kortsarz, Guy

doi:10.1007/bfb0053970

Cited by 44 publications

(74 citation statements)

References 20 publications

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“…client-server 2-spanner [21] and fault-tolerant 2-spanner [18], [19] (for which only O(log ∆) is known). All of these bounds are basically optimal, assuming P = NP, due to a hardness result of Kortsarz [28].…”

Section: Related Workmentioning

confidence: 99%

Everywhere-Sparse Spanners via Dense Subgraphs

E¹,

Dinitz

Krauthgamer

2012

2012 IEEE 53rd Annual Symposium on Foundations of Computer Science

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Abstract-The significant progressg in constructing graph spanners that are sparse (small number of edges) or light (low total weight) has skipped spanners that are everywhere-sparse (small maximum degree). This disparity is in line with other network design problems, where the maximum-degree objective has been a notorious technical challenge. Our main result is for the LOWEST DEGREE 2-SPANNER (LD2S) problem, where the goal is to compute a 2-spanner of an input graph so as to minimize the maximum degree. We design a polynomial-time algorithm achieving approximation factorÕ(∆where ∆ is the maximum degree of the input graph. The previous O(∆ 1/4 )-approximation was proved nearly two decades ago by Kortsarz and Peleg [SODA 1994, SICOMP 1998].Our main conceptual contribution is to establish a formal connection between LD2S and a variant of the DENSEST k-SUBGRAPH (DkS) problem. Specifically, we design for both problems strong relaxations based on the Sherali-Adams linear programming (LP) hierarchy, and show that "faithful" randomized rounding of the DkS-variant can be used to round LD2S solutions. Our notion of faithfulness intuitively means that all vertices and edges are chosen with probability proportional to their LP value, but the precise formulation is more subtle.Unfortunately, the best algorithms known for DkS use the Lovász-Schrijver LP hierarchy in a non-faithful way [Bhaskara, Charikar, Chlamtac, Feige, and Vijayaraghavan, STOC 2010]. Our main technical contribution is to overcome this shortcoming, while still matching the gap that arises in random graphs by planting a subgraph with same log-density.

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Section: Related Workmentioning

confidence: 99%

Everywhere-Sparse Spanners via Dense Subgraphs

E¹,

Dinitz

Krauthgamer

2012

2012 IEEE 53rd Annual Symposium on Foundations of Computer Science

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“…For completeness, we note that the basic k-spanner problem has been solved for the special case of k = 2: there is an O(log n) approximation [25], and that is the best possible [24].…”

Section: The Basic K-spanner Problem and Previous Workmentioning

confidence: 99%

“…The standard reduction from Label Cover to Min-Rep [24] entails duplications of many super vertices. This is needed in order to ensure regularity in the Min-Rep graph, which is used to ensure that removing a µ fraction of the supervertices will imply a removal of at most a µ fraction of the superedges.…”

Section: Some Remarks On Our Techniquesmentioning

confidence: 99%

Label Cover Instances with Large Girth and the Hardness of Approximating Basic k-Spanner

Dinitz

Kortsarz

Raz

2012

Automata, Languages, and Programming

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We study the well-known Label Cover problem under the additional requirement that problem instances have large girth. We show that if the girth is some k, the problem is roughly 2 log 1− n/k hard to approximate for all constant > 0. A similar theorem was claimed by Elkin and Peleg [ICALP 2000], but their proof was later found to have a fundamental error. We use the new proof to show inapproximability for the basic k-spanner problem, which is both the simplest problem in graph spanners and one of the few for which super-logarithmic hardness was not known. Assuming N P ⊆ BP T IM E(2 polylog(n) ), we show that for every k ≥ 3 and every constant > 0 it is hard to approximate the basic k-spanner problem within a factor better than 2 (log 1− n)/k (for large enough n). A similar hardness for basic k-spanner was claimed by Elkin and Peleg [ICALP 2000], but the error in their analysis of Label Cover made this proof fail as well. Thus for the problem of Label Cover with large girth we give the first non-trivial lower bound. For the basic k-spanner problem we improve the previous best lower bound of Ω(log n)/k from [24]. Our main technique is subsampling the edges of 2-query PCPs, which allows us to reduce the degree of a PCP to be essentially equal to the soundness desired. This turns out to be enough to essentially guarantee large girth.

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“…In the MinRep [24] problem we are given a bipartite graph G = (A, B, E) with a partition of A and B into equal-sized subsets. Let q A and q B denote the number of sets in the partition of A and B, respectively.…”

Section: The Minrep Problemmentioning

confidence: 99%

Approximation Algorithms and Hardness for Domination with Propagation

Aazami¹,

Stilp²

2009

SIAM J. Discrete Math.

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The power dominating set (PDS) problem is the following extension of the well-known dominating set problem: find a smallest-size set of nodes S that power dominates all the nodes, where a node v is power dominated if (1) v is in S or v has a neighbor in S, or (2) v has a neighbor w such that w and all of its neighbors except v are power dominated. We show a hardness of approximation threshold of 2 log 1−ǫ n in contrast to the logarithmic hardness for the dominating set problem. We give an O( √ n) approximation algorithm for planar graphs, and show that our methods cannot improve on this approximation guarantee. Finally, we initiate the study of PDS on directed graphs, and show the same hardness threshold of 2 log 1−ǫ n for directed acyclic graphs. Also we show that the directed PDS problem can be solved optimally in linear time if the underlying undirected graph has bounded tree-width.

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On the hardness of approximating spanners

Cited by 44 publications

References 20 publications

Everywhere-Sparse Spanners via Dense Subgraphs

Everywhere-Sparse Spanners via Dense Subgraphs

Label Cover Instances with Large Girth and the Hardness of Approximating Basic k-Spanner

Approximation Algorithms and Hardness for Domination with Propagation

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