Download presentation

Presentation is loading. Please wait.

Published byMaya Barr Modified over 3 years ago

1
Approximate Max-integral-flow/min-cut Theorems Kenji Obata UC Berkeley June 15, 2004

2
Multicommodity Flow Graph G, edge capacities c, demands K

3
Multicommodity Flow K-partition

4
Multicommodity Flow K-cut

5
Multicommodity Flow

6
… for one commodity [Ford-Fulkerson] Multicommodity Flow

7
… in general [Leighton-Rao, GVY] Multicommodity Flow

8
Integral Multicommodity Flow Suppose c is integral. Can we find integral f ? … for one commodity, yes [Ford-Fulkerson] … in general, no [Garg] Both flow [GVY] and cut [DJPSY] problems are NP- hard

9
Integral Multicommodity Flow Suppose every K-cut has weight >= C. (this work)

10
Integral Multicommodity Flow Suppose every K-cut has weight >= C. Theorem: For any G, R = O( -1 log k) If G is planar, R = O( -1 ) If G is -dense, R = O( -1/2 -1/2 ) (this work)

11
Integral Multicommodity Flow Algorithmic: Construct an integral flow or a proof that the K-cut condition is violated => edge-disjoint path problems => odd circuit cover problems => property testing (this work)

12
Algorithm (general graphs) Greed g(t) Time t (not to scale)

13
Algorithm (general graphs) Greed g(t) Time t (not to scale)

14
Algorithm (special cases) Greed g(t) Time t (not to scale) planar dense

15
Constructing g(t)

18
Bounding f( ) General graphs Reinterpret [GVY] applied to original graph metric (Note: Makes no sense) Planar graphs … [Klein-Plotkin-Rao] Dense graphs

19
Bounding f( ) (dense case) |E(G)| >= n 2, > 0, c {0,1} E B(v, ) = ball of radius around v, boundary B o (v, ) B(v, ) B o (v, )

20
Choose arbitrary vertex v, set = 0 While |B o (v, )| |B o (v, )| > |B(v, )| |B(v, )|, grow Bounding f( ) (dense case) B(v, ) B o (x, +1)

21
Bounding f( ) (dense case) Each ball has low radius Proof:

22
Bounding f( ) (dense case) Induced multicut has low density Proof: Together (set ) =>

23
Proof of Theorem Suppose every K-cut has weight >= C Claim: K-path of length <= g( ):

24
Proof of Theorem

25
Proof of Theorem (contd) Delete path p (|p| <= g( )) and iterate c = c – p ; = – p/C Witness for flow f, residual multicut m

26
Edge-disjoint paths Corollary: If G has degree bound, min-multicut m then

27
Motivation (Property Testing) Given bounded degree graph G Want to distinguish whether G has a certain property or is far ( n entries) from having the property In sub-linear (constant?) time Example: Coloring problems No sub-linear algorithms for 3-coloring [BOT] 2-coloring has complexity ~O(n 1/2 )

28
Testing 2-Colorability Fix max-cut Set G = {crossing edges}, K = {internal edges} => min-multicut has weight >= m

29
By corollary, -2 m edge-disjoint odd cycles of length O( -2 ) Algorithm: Repeat O (log (1/ )) times: Sample random vertex v Do BFS about v to depth 1/ With probability 1-, find odd cycle using exp( O( -2 )) log( -1 ) queries Testing 2-Colorability (planar case)

30
Thank you

Similar presentations

OK

CHAPTER SIX T HE P ROBABILISTIC M ETHOD M1 Zhang Cong 2011/Nov/28.

CHAPTER SIX T HE P ROBABILISTIC M ETHOD M1 Zhang Cong 2011/Nov/28.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt online viewership Ppt on thermal power plant Ppt on natural disasters Ppt on p&g products philippines Converter pub to ppt online templates Ppt on two stage rc coupled amplifier Ppt on musical instruments in hindi Ppt on preservation methods of food Ppt on astronomy and astrophysics videos Ppt on programmable logic array programmer