ℓ 2 2 Spreading Metrics for Vertex Ordering Problems

We show that for every n-point metric space M and positive integer k, there exists a spanning tree T with unweighted diameter O(k) and weight, and a spanning tree T ′ with weight w(T ′ ) = O(k) · w(M ST (M )) and unweighted diameter O(k · n 1/k ). These trees also achieve an optimal maximum degree. Furthermore, we demonstrate that these trees can be constructed efficiently.We prove that these tradeoffs between unweighted diameter and weight are tight up to constant factors in the entire range of parameters. Moreover, our lower bounds apply to a basic 1-dimensional Euclidean space.Our lower bounds for the particular case of unweighted diameter O(log n) are of independent interest, settling a long-standing open problem in Computational Geometry. In STOC'95 Arya et al. devised a construction of Euclidean Spanners with unweighted diameter O(log n) and weight O(log n) · w(M ST (M )). In SODA'05 Agarwal et al. showed that this result is tight up to a factor of O(log log n). We close this gap and show that the result of Arya et al. is tight up to constant factors.Finally, our upper bounds imply improved approximation algorithms for the minimum h-hop spanning tree and bounded diameter minimum spanning tree problems for metric spaces.

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“…There is an extensive literature on the minimum linear arrangement problem itself. (See [20,33], and the references therein).…”

Section: Related Workmentioning

confidence: 99%

Low-Light Trees, and Tight Lower Bounds for Euclidean Spanners

Dinitz

Elkin

Solomon

2009

Discrete Comput Geom

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“…See, for example, the work of Charikar et al (2010). The current best approximation algorithm for IGC achieves a ratio of O( √ log n log log n) (Charikar et al, 2010).…”

Section: The Connection: Layout Problemsmentioning

confidence: 99%

“…The current best approximation algorithm for IGC achieves a ratio of O( √ log n log log n) (Charikar et al, 2010). It turns out that the SSE Conjecture can be used to prove super-constant hardness for this problem as well.…”

Section: The Connection: Layout Problemsmentioning

confidence: 99%

“…In the groundbreaking work of Arora et al (2009), semidefinite programming was used to give an improved approximation ratio of O( √ log n) for c-balanced separator. Using their ideas, improved algorithms for ordering problems have been found, such as the O( √ log n log log n) approximation algorithm for IGC and MLA (Charikar et al, 2010), the O( √ log n) approximation algorithm for Treewidth (Feige et al, 2005) and the O( √ log n log n) approximation algorithm for Pathwidth (Feige et al, 2005).…”

Section: Previous Workmentioning

confidence: 99%

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Inapproximability of Treewidth and Related Problems

Austrin

Pitassi

et al. 2014

jair

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Graphical models, such as Bayesian Networks and Markov networks play an important role in artificial intelligence and machine learning. Inference is a central problem to be solved on these networks. This, and other problems on these graph models are often known to be hard to solve in general, but tractable on graphs with bounded Treewidth. Therefore, finding or approximating the Treewidth of a graph is a fundamental problem related to inference in graphical models. In this paper, we study the approximability of a number of graph problems: Treewidth and Pathwidth of graphs, Minimum Fill-In, OneShot Black (and Black-White) pebbling costs of directed acyclic graphs, and a variety of different graph layout problems such as Minimum Cut Linear Arrangement and Interval Graph Completion. We show that, assuming the recently introduced Small Set Expansion Conjecture, all of these problems are NP-hard to approximate to within any constant factor in polynomial time.

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“…In the one-dimensional case (y max = 1) this is the famous optimum linear arrangement problem, for which an O( log |V (G)| log log |V (G)|)-factor approximation algorithm was found by Charikar et al [2006] and Feige and Lee [2007]. However, the optimum linear arrangement problem is not known to be MAXSNP-hard (but see Ambühl, Mastrolilli and Svensson [2007]).…”

Section: What Is Knownmentioning

confidence: 99%

Combinatorial Problems in Chip Design

Körte

Vygen

2008

Bolyai Society Mathematical Studies

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The design of very large scale integrated (VLSI) chips is an exciting area of applying mathematics, posing constantly new challenges.We present some important and challenging open problems in various areas of chip design. Although the problems are motivated by chip design, they are formulated mathematically; understanding and solving them does not require any knowledge of chip design. We give some partial results and argue why a full resolution of one of the problems could result in an advance of the state of the art in algorithms for chip design.

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ℓ 2 2 Spreading Metrics for Vertex Ordering Problems

Cited by 38 publications

References 32 publications

Low-Light Trees, and Tight Lower Bounds for Euclidean Spanners

Low-Light Trees, and Tight Lower Bounds for Euclidean Spanners

Inapproximability of Treewidth and Related Problems

Combinatorial Problems in Chip Design

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