1. Approximation Algorithms for Problems
on Networks and Streams of Data
Luca Foschini - Ph.D. Defense
Committee: Subhash Suri (chair), John Gilbert, Teofilo Gonzalez

2. Why Approximation Algorithms?

3. Why Approximation Algorithms?
Exact algorithms require many resources

4. Why Approximation Algorithms?
Hardware
Exact algorithms require many resources
Apps
Data

5. Why Approximation Algorithms?
Hardware
Exact algorithms require many resources
Apps
Problems solvable
exactly Data

6. A Long History,
and Work in Progress
© Original Artist

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 Artist

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 needed

9. Content of the Dissertation

10. Content of the Dissertation
"

11. Content of the Dissertation
Networks
"
Data Streams

12. Content of the Dissertation
STACS12 +
Partitioning
Algorithmica
Networks
SODA11 +
Shortest Paths
Algorithmica
"
Time Series ICDE10
Data Streams
Burst Detection NSDI11

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 WAW09

14. Roadmap
STACS12 +
Partitioning
Algorithmica
Networks
SODA11 +
Shortest Paths
Algorithmica
"
Time Series ICDE10
Data Streams
Burst Detection NSDI11

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)

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]

17. Related Work

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]

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?

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 = 8

21. Trees Are Hard

22. Trees Are Hard
✤ NP-hard to approx. cut size for !=nc
(for any c<1) even if constant diameter

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 degree

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 trees

25. Relax!

26. Relax!
Balance constraint relaxed:
|Vi |  (1 + ")dn/ke

27. Relax!
Balance constraint relaxed:
|Vi |  (1 + ")dn/ke
Balance relaxed
Perfect balance
Optimal cut size
Cut size
approximated
!

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
!

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?

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/") " " )

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 ! =1

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) improves

33. Tree Decomposition

34. Analysis of Embedding

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 )

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 Trees

37. Roadmap
STACS12 +
Partitioning
Algorithmica
Networks
SODA11 +
Shortest Paths
Algorithmica
"
Time Series ICDE10
Data Streams
Burst Detection NSDI11

38. Approximating Time Series
✤ Represent a time series with B
linear segments
✤ New value arrives to the time
series, need to reallocate
segments

39. Approximating Time Series
✤ Represent a time series with B
linear segments
✤ New value arrives to the time
series, need to reallocate
segments

40. Approximating Time Series
✤ Represent a time series with B
linear segments
✤ New value arrives to the time
series, need to reallocate
segments

41. Old Algorithms, New Proofs

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)

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)

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 norms

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 algorithms