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Approximation Algorithms for Problems
on Networks and Streams of Data
Luca Foschini - Ph.D. Defense

Committee: Subhash Suri (chair), John Gilbert, Teoﬁlo Gonzalez

Friday, September 7, 12

Why Approximation Algorithms?



Exact algorithms require many resources


Hardware

Apps

Data


Hardware

Apps

Problems solvable
exactly Data


A Long History,
and Work in Progress

© Original Artist


A Long History,

✤ Early ‘70s - many combinatorial
problems found to be NP-hard

✤ Recently - more restricting
computation models proposed e.g.,
data stream

© Original Artist


A Long History,

✤ 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


Content of the Dissertation



"



Networks

"

Data Streams



STACS12 +
Partitioning
Algorithmica
Networks
SODA11 +
Shortest Paths
Algorithmica
"
Time Series ICDE10

Data Streams

Burst Detection NSDI11



STACS12 + ICISS08
Partitioning
Algorithmica
Networks ICIP11
SODA11 +
Shortest Paths ALENEX10
Algorithmica
"
ESA11
Time Series ICDE10

Data Streams WOOT11

Burst Detection NSDI11 WAW09


Roadmap

STACS12 +
Partitioning
Algorithmica
Networks
SODA11 +
Shortest Paths
Algorithmica
"
Time Series ICDE10

Data Streams

Burst Detection NSDI11


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)

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]


Related Work


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]




✤ NP-hard to approximate cut size
Räcke 2006]

Trees - simple instances?




✤ NP-hard to approximate cut size n=31, k=8 cut size = 10
Räcke 2006]

Trees - simple instances?

n=31, k=9 cut size = 8

Trees Are Hard


Trees Are Hard

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


Trees Are Hard


✤ APX-hard to approx. cut-size even if
constant degree


Trees Are Hard


✤ APX-hard to approx. cut-size even if
constant degree

Most NP-hard problems become trivial on trees


Relax!


Relax!

Balance constraint relaxed:
|Vi |  (1 + ")dn/ke


Relax!

Balance constraint relaxed:
|Vi |  (1 + ")dn/ke

Balance relaxed
Perfect balance
Optimal cut size
Cut size
approximated
!


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
!


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?


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/") " " )


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


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


Tree Decomposition


Analysis of Embedding


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 )


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

Approximating Time Series

✤ Represent a time series with B
linear segments

✤ New value arrives to the time
series, need to reallocate
segments


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)




✤ 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

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


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