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Approximation for Directed Spanner

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This paper presents an efficient approximation algorithm for the Directed Spanner problem, focusing on K-spanners that maintain distances up to a factor of k > 1 in directed graphs. It highlights the significance of directed spanners in various applications, including efficient routing and simulating synchronized protocols in unsynchronized networks. Key concepts include antispanners, which identify edges that disrupt paths, and innovative techniques such as linear programming relaxation and randomized rounding. The research aims to improve approximation factors while addressing the challenges of quasi-NP-hardness associated with this problem.

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Approximation for Directed Spanner

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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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