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Efficient Approximation Algorithms for Directed Spanners: Enhancements and Applications

This presentation discusses the Directed Spanner Problem, aiming to find the sparsest k-spanner of a directed graph while preserving distances within a stretch factor k>1. It highlights the significance of spanners in various applications, such as efficient routing, simulating synchronized protocols, and developing algorithms for distance oracles. Key concepts include antispanners and the linear programming relaxation method, along with techniques for improving approximation factors. The work is based on a collaborative paper presented at ICALP'11, involving prominent researchers and addressing critical open questions.

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Efficient Approximation Algorithms for Directed Spanners: Enhancements and Applications

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  1. Approximation for Directed Spanner GrigoryYaroslavtsev Penn State + AT&T Labs (intern) Based on a paper at ICALP’11, joint with Berman (PSU), Bhattacharyya (MIT), Makarychev (IBM), Raskhodnikova (PSU)

  2. Directed Spanner Problem • K-Spanner- subset of edges, preserving distances up to a factor k > 1 (stretch k). • Problem:Find the sparsest k-spanner of a directed graph.

  3. Why Spanners? • Applications • Efficient routing • Simulating synchronized protocols in unsynchronized networks • Parallel/Distributed/Streaming approximation algorithms for shortest paths • Algorithms for distance oracles

  4. Why Directed Spanners? • Property testing and property reconstruction • Techniques give improvement for: • Directed Steiner Forest • Min-cost Directed Spanner (improvement for unit-length, constant k), Unit-lengthDirected Spanners, Client-server k-Spanner, K-diameterSpanning Subgraph, … • Practical under natural assumptions

  5. Previous work Similar problems considered in • [Dodis, Khanna, STOC 99] • [Feldman, Kortsarz, Nutov, SODA 09] • [Bhattacharyya, Grigorescu, Jung, Raskhodnikova, Woodruff, SODA 09] • [Berman, Raskhodnikova, Ruan, FSTTCS 10] • … • [Dinitz, Krauthgamer, STOC 11]

  6. Key idea: Antispanners • Antispanner – subset of edges, which destroys all paths from A to B of stretch at most k. • We have a spanner if and only if we hit all minimalantispanners

  7. Linear programming relaxation subject to: for all minimalantispannersA for all edges. • How to solve the LP? • # of antispannersis exponential in n =>separation oracle. • Randomized rounding • Counting minimal antispanners (technical part)

  8. Approximation factor • Antispanners are in local graphs • Sampling [BGJRW09] => reduce size of local graphs • # of antispanners is exponential in size of local graph • Õ(-approximation • Previous: [DK11]

  9. Directed Spanner Problem (recap) • K-Spanner- subset of edges, preserving distances up to a factor k > 1 (stretch k). • Problem:Find the sparsest k-spanner of a directed graph.

  10. What do wethink is next • Improve approximation factor • Currently: Õ(-approximation • Hardness: Quasi-NP-hardness, [Elkin, Peleg, STOC 00] • Integrality gap: Ω[DK11]. • Improve counting? Other techniques? • … • What is a natural online setting?

  11. What do you think?

  12. Thank you! • More information: http://grigory.us

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