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  • 1. Approximation Algorithms for Problems on Networks and Streams of Data Luca Foschini - Ph.D. Defense Committee: Subhash Suri (chair), John Gilbert, Teofilo GonzalezFriday, September 7, 12
  • 2. Why Approximation Algorithms?Friday, September 7, 12
  • 3. Why Approximation Algorithms? Exact algorithms require many resourcesFriday, September 7, 12
  • 4. Why Approximation Algorithms? Hardware Exact algorithms require many resources Apps DataFriday, September 7, 12
  • 5. Why Approximation Algorithms? Hardware Exact algorithms require many resources Apps Problems solvable exactly DataFriday, September 7, 12
  • 6. A Long History, and Work in Progress © Original ArtistFriday, September 7, 12
  • 7. A Long History, and Work in Progress ✤ Early ‘70s - many combinatorial problems found to be NP-hard ✤ Recently - more restricting computation models proposed e.g., data stream © Original ArtistFriday, September 7, 12
  • 8. A Long History, and Work in Progress ✤ Early ‘70s - many combinatorial problems found to be NP-hard ✤ Recently - more restricting computation models proposed e.g., data stream © Original Artist Heuristics not sufficient, provable guarantees neededFriday, September 7, 12
  • 9. Content of the DissertationFriday, September 7, 12
  • 10. Content of the Dissertation "Friday, September 7, 12
  • 11. Content of the Dissertation Networks " Data StreamsFriday, September 7, 12
  • 12. Content of the Dissertation STACS12 + Partitioning Algorithmica Networks SODA11 + Shortest Paths Algorithmica " Time Series ICDE10 Data Streams Burst Detection NSDI11Friday, September 7, 12
  • 13. Content of the Dissertation STACS12 + ICISS08 Partitioning Algorithmica Networks ICIP11 SODA11 + Shortest Paths ALENEX10 Algorithmica " ESA11 Time Series ICDE10 Data Streams WOOT11 Burst Detection NSDI11 WAW09Friday, September 7, 12
  • 14. Roadmap STACS12 + Partitioning Algorithmica Networks SODA11 + Shortest Paths Algorithmica " Time Series ICDE10 Data Streams Burst Detection NSDI11Friday, September 7, 12
  • 15. k-Balanced Partitioning Problem Given: an unweighted graph G on n vertices; an integer k Find: a partition of the vertices of G into k sets Vi s.t. ✤ |Vi |  dn/ke ✤ Cut size (number of edges connecting vertices in different Vi) is minimized joint work with Andi Feldmann (ETHz) (appeared in STACS12, submitted to Algorithmica)Friday, September 7, 12
  • 16. Motivation & Complexity ✤ Divide-and-conquer algorithms ✤ VLSI design ✤ Parallel computing ✤ NP-hard to approximate cut size within any finite value alpha [Andreev and Räcke 2006]Friday, September 7, 12
  • 17. Related WorkFriday, September 7, 12
  • 18. General Graphs & Trees ✤ Algorithm is !-approximation if finds a cut at most ! times optimal ✤ NP-hard to approximate cut size within any finite ! [Andreev and Räcke 2006]Friday, September 7, 12
  • 19. General Graphs & Trees ✤ Algorithm is !-approximation if finds a cut at most ! times optimal ✤ NP-hard to approximate cut size within any finite ! [Andreev and Räcke 2006] Trees - simple instances?Friday, September 7, 12
  • 20. General Graphs & Trees ✤ Algorithm is !-approximation if finds a cut at most ! times optimal ✤ NP-hard to approximate cut size n=31, k=8 cut size = 10 within any finite ! [Andreev and Räcke 2006] Trees - simple instances? n=31, k=9 cut size = 8Friday, September 7, 12
  • 21. Trees Are HardFriday, September 7, 12
  • 22. Trees Are Hard ✤ NP-hard to approx. cut size for !=nc (for any c<1) even if constant diameterFriday, September 7, 12
  • 23. Trees Are Hard ✤ NP-hard to approx. cut size for !=nc (for any c<1) even if constant diameter ✤ APX-hard to approx. cut-size even if constant degreeFriday, September 7, 12
  • 24. Trees Are Hard ✤ NP-hard to approx. cut size for !=nc (for any c<1) even if constant diameter ✤ APX-hard to approx. cut-size even if constant degree Most NP-hard problems become trivial on treesFriday, September 7, 12
  • 25. Relax!Friday, September 7, 12
  • 26. Relax! Balance constraint relaxed: |Vi |  (1 + ")dn/keFriday, September 7, 12
  • 27. Relax! Balance constraint relaxed: |Vi |  (1 + ")dn/ke Balance relaxed Perfect balance Optimal cut size Cut size approximated !Friday, September 7, 12
  • 28. Relax! Balance constraint relaxed: Bicriteria Approximation: cut size approximation ! measured |Vi |  (1 + ")dn/ke w.r.t perfectly balanced optimum Balance relaxed Perfect balance Optimal cut size Cut size approximated !Friday, September 7, 12
  • 29. 0<eps<1 on general graphs ✤ eps>1 -- alpha in .... spreading metric techniques ✤ 0<eps < 1 not much improvement. 1/epsˆ2 log ^1.5 n ✤ What about trees?Friday, September 7, 12
  • 30. Summary of PTAS for Trees ✤ Compute optimal cut size for each coarse signature using DP ✤ Pack each coarse signatures into bins of size (1 + ")dn/ke ✤ Pick solution with smallest cut size among those fitting into k bins 4 1+3d 1 log( 1 )e ✤ Total time complexity O(n (k/") " " )Friday, September 7, 12
  • 31. Summary of PTAS for Trees ✤ Compute optimal cut size for each coarse signature using DP ✤ Pack each coarse signatures into bins of size (1 + ")dn/ke ✤ Pick solution with smallest cut size among those fitting into k bins 4 1+3d 1 log( 1 )e ✤ Total time complexity O(n (k/") " " ) Show that ! =1Friday, September 7, 12
  • 32. Extension to General Graphs ✤ Decomposition of graph into collection of trees [Räcke, Madry], cut size worsen by at most O(log n) for at least 1 tree ✤ Apply PTAS for trees to each instance ✤ Return partition for tree with minimum cut ✤ alpha = O(log n) improvesFriday, September 7, 12
  • 33. Tree DecompositionFriday, September 7, 12
  • 34. Analysis of EmbeddingFriday, September 7, 12
  • 35. Extensions & Open Problems ✤ Tree embedding techniques allow the !=1 tree PTAS to translate to a !=O(log n) approx for general weighted graphs ✤ Improves on previous best != O(log 1.5 n/"2 )Friday, September 7, 12
  • 36. Extensions & Open Problems ✤ Tree embedding techniques allow the !=1 tree PTAS to translate to a !=O(log n) approx for general weighted graphs ✤ Improves on previous best != O(log 1.5 n/"2 )    Graphs TreesFriday, September 7, 12
  • 37. Roadmap STACS12 + Partitioning Algorithmica Networks SODA11 + Shortest Paths Algorithmica " Time Series ICDE10 Data Streams Burst Detection NSDI11Friday, September 7, 12
  • 38. Approximating Time Series ✤ Represent a time series with B linear segments ✤ New value arrives to the time series, need to reallocate segmentsFriday, September 7, 12
  • 39. Approximating Time Series ✤ Represent a time series with B linear segments ✤ New value arrives to the time series, need to reallocate segmentsFriday, September 7, 12
  • 40. Approximating Time Series ✤ Represent a time series with B linear segments ✤ New value arrives to the time series, need to reallocate segmentsFriday, September 7, 12
  • 41. Old Algorithms, New ProofsFriday, September 7, 12
  • 42. Old Algorithms, New Proofs ✤ We prove that a popular greedy merge scheme gives constant (bicriteria) approx. for many L_p norms. (ICDE10; joint with Gandhi, Suri)Friday, September 7, 12
  • 43. Old Algorithms, New Proofs ✤ We prove that a popular greedy merge scheme gives constant (bicriteria) approx. for many L_p norms. (ICDE10; joint with Gandhi, Suri) ✤ Results implemented in Linux Kernel and used to detect traffic bursts in networks (NSDI11, joint with Uyeda, Suri, Varghese, Baker)Friday, September 7, 12
  • 44. Old Algorithms, New Proofs ✤ We prove that a popular greedy merge scheme gives constant (bicriteria) approx. for many L_p norms. (ICDE10; joint with Gandhi, Suri) ✤ Results implemented in Linux Kernel and used to detect traffic bursts in networks (NSDI11, joint with Uyeda, Suri, Varghese, Baker) Next steps: Extend results in ICDE10 to other normsFriday, September 7, 12
  • 45. Conclusion ✤ Approximation is necessary to reduce resource utilization ✤ Presented approximation algorithms for problems from different domains that we cannot afford to solve exactly ✤ Presented basic building blocks that can be used across the board to design approximation algorithmsFriday, September 7, 12