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