1 / 30

Approximate Max-integral-flow

ulf
Download Presentation

Approximate Max-integral-flow

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

    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. Multicommodity Flow

    7. Multicommodity Flow Interesting because flow problem is poly computable, while cut problem is NP-hard => approximation algorithms for cut problemsInteresting because flow problem is poly computable, while cut problem is NP-hard => approximation algorithms for cut problems

    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 Beautiful property of Ford-Fulkerson Thm is integral capacities => integral flow. Unlike fractional case, no analogous equality nor approximation. Also, *both* flow and cut problems are NP-hard.Beautiful property of Ford-Fulkerson Thm is integral capacities => integral flow. Unlike fractional case, no analogous equality nor approximation. Also, *both* flow and cut problems are NP-hard.

    9. Integral Multicommodity Flow Takeaway: This turns out to be an extremely natural parametrization of multicommodity flow problems.Takeaway: This turns out to be an extremely natural parametrization of multicommodity flow problems.

    10. Integral Multicommodity Flow Dense graph has unit capacities.Dense graph has unit capacities.

    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 In other words, when multicut is dense, not only does min-cut become a good approximation to max-integral-flow, but it becomes easy to do integral routing between terminals.In other words, when multicut is dense, not only does min-cut become a good approximation to max-integral-flow, but it becomes easy to do integral routing between terminals.

    12. Algorithm (general graphs)

    13. Algorithm (general graphs)

    15. Constructing g(t) Radius refers to # edges (not weighted) G some family of graphsRadius refers to # edges (not weighted) G some family of graphs

    16. Constructing g(t)

    17. Constructing g(t)

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

    19. Bounding f(r) (dense case) |E(G)| >= dn2, d > 0, c {0,1}E B(v, r) = ball of radius r around v, boundary Bo(v, r)

    20. Choose arbitrary vertex v, set r = 0 While |Bo(v, r)| |Bo(v, r+1)| > a |B(v, )| |B(v, r)|, grow Bounding f(r) (dense case)

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

    22. Bounding f(r) (dense case) Induced multicut has low density Proof:

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

    24. Proof of Theorem

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

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

    27. Motivation (Property Testing) Given bounded degree graph G Want to distinguish whether G has a certain property or is far (en 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(n1/2)

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

    29. Testing 2-Colorability (planar case)

    30. Thank you

More Related