ApproximateMax-integral-flow/min-cut Theorems

Kenji Obata

UC Berkeley

June 15, 2004

Multicommodity Flow

Graph G, edge capacities c, demands K

Multicommodity Flow

K-partition

Multicommodity Flow

K-cut

Multicommodity Flow

… for one commodity [Ford-Fulkerson]

Multicommodity Flow

… in general [Leighton-Rao, GVY]

Multicommodity Flow

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

Integral Multicommodity Flow

Suppose every K-cut has weight >= C.

(this work)

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)

Integral Multicommodity Flow

Algorithmic:Construct an integral flowor a proof that the K-cut condition is violated

=> edge-disjoint path problems => odd circuit cover problems => property testing

(this work)

Algorithm (general graphs)

Gre

ed g

(t)

Time t(not to scale)

Algorithm (general graphs)

Gre

ed g

(t)

Time t(not to scale)

Algorithm (special cases)

Gre

ed g

(t)

Time t(not to scale)

planar

dense

Constructing g(t)

radius of partitions-KKP

Constructing g(t)

)),...,((min

1max)( 1

),...,(,

1t

SSkKck SSc

Cf

Kt

P G,

G

radius of partitions-KKP

Constructing g(t)

)),...,((min

1max)( 1

),...,(,

1t

SSkKck SSc

Cf

Kt

P G,

G

2min))((

)( f

tg

radius of partitions-KKP

Bounding f()

General graphsReinterpret [GVY] applied to original graph metric(Note: Makes no sense)

Planar graphs… [Klein-Plotkin-Rao]

Dense graphs

Bounding f() (dense case)

|E(G)| >= n2, > 0, c {0,1}E

B(v, ) = ball of radius around v, boundary Bo(v, )

B(v, )

Bo(v, )

Choose arbitrary vertex v, set = 0 While |Bo(v, )| |Bo(v, )| > |B(v, )| |B(v, )|, grow

Bounding f() (dense case)

B(v, )

Bo(x, +1)

Bounding f() (dense case)

Each ball has low radius

Proof:

nB

nB

nB

BBnBBBnB

nBbb

vBBvBbnBbb

1449'

' ''' ; '

least at , of oneleast At

,(,),(

120

o

Bounding f() (dense case)

Induced multicut has low density

Proof:

Cn

SnnS

ScSSc

ii

ii

iit

2

1 ))(()),...,((

Together (set ) =>

144)(f

Proof of Theorem

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

2min))((

)( f

tg

Proof of Theorem

2min))((

)( f

tg

)),...,((min

1max)( 1

),...,(,

1t

SSkKck SSc

Cf

Kt

P G,

G

Proof of Theorem (cont’d)

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

CdttgFfwF

0

))( s.t. max)(

)()]([: ecemfEeM

Edge-disjoint paths

Corollary:

If G has degree bound , min-multicut m then

paths. graphs, dense For

paths. graphs, planar For

paths disjoint-edgemutually

m

m

mk

log

Motivation (Property Testing)

Given bounded degree graph G Want to distinguish whether

G has a certain propertyor is far (n entries) from having the propertyIn sub-linear (constant?) time

Example: Coloring problemsNo sub-linear algorithms for 3-coloring [BOT]

2-coloring has complexity ~O(n1/2)

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

By corollary, -2 m edge-disjoint odd cycles of length O(-2)

Algorithm:Repeat O (log (1/)) times:Sample random vertex vDo BFS about v to depth 1/

With probability 1-, find odd cycle usingexp(O(-2)) log(-1) queries

Testing 2-Colorability (planar case)

Thank you