Approximate Maxintegralflowmincut Theorems

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ApproximateMax-integral-flow/min-cut Theorems

• Kenji Obata
• UC Berkeley
• June 15, 2004

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Multicommodity Flow

• Graph G, edge capacities c, demands K

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Multicommodity Flow

• K-partition

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Multicommodity Flow

• K-cut

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Multicommodity Flow
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Multicommodity Flow
for one commodity Ford-Fulkerson
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Multicommodity Flow
in general Leighton-Rao, GVY
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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

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Integral Multicommodity Flow
(this work)
Suppose every K-cut has weight gt eC.
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Integral Multicommodity Flow
(this work)

• Suppose every K-cut has weight gt eC.
• Theorem
• For any G, R O(e-1 log k)
• If G is planar, R O(e-1)
• If G is d-dense, R O(e-1/2 d-1/2)

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Integral Multicommodity Flow
(this work)

• Algorithmic
• Construct an integral flow
• or a proof that the K-cut condition is violated
• gt edge-disjoint path problems
• gt odd circuit cover problems
• gt property testing

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Algorithm (general graphs)
Greed g(t)
Time t
(not to scale)
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Algorithm (general graphs)
Greed g(t)
Time t
(not to scale)
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Algorithm (special cases)
planar
dense
Greed g(t)
Time t
(not to scale)
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Constructing g(t)
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Constructing g(t)
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Constructing g(t)
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Bounding f(r)

• General graphs
• Reinterpret GVY applied to original graph
  metric
• (Note Makes no sense)
• Planar graphs
• Klein-Plotkin-Rao
• Dense graphs

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Bounding f(r) (dense case)

• E(G) gt dn2, d gt 0, c Î 0,1E
• B(v, r) ball of radius r around v, boundary
  Bo(v, r)

r
Bo(v, r)
B(v, r)
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Bounding f(r) (dense case)

• Choose arbitrary vertex v, set r 0
• While Bo(v, r) Bo(v, r1) gt a B(v, ) B(v,
  r), grow

Bo(x, r1)
r
B(v, r)
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Bounding f(r) (dense case)

• Each ball has low radius
• Proof

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Bounding f(r) (dense case)

• Induced multicut has low density
• Proof

• Together (set a) gt

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Proof of Theorem

• Suppose every K-cut has weight gt eC
• Claim K-path of length lt g(e)

2r
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Proof of Theorem

2r
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Proof of Theorem (contd)

• Delete path p (p lt g(e)) and iterate
• c c p e e p/C
• Witness for flow f, residual multicut m

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Edge-disjoint paths

• Corollary
• If G has degree bound D, min-multicut em then

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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)

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Testing 2-Colorability

• Fix max-cut
• Set G crossing edges, K internal edges
• gt min-multicut has weight gt em

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Testing 2-Colorability (planar case)

• By corollary, W (e-2 m) edge-disjoint odd
  cycles of length O(e-2)
• Algorithm
• Repeat O (log (1/d)) times
• Sample random vertex v
• Do BFS about v to depth 1/e2
• With probability 1-d, find odd cycle
  usingexp(O(e-2)) log(d-1) queries

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Thank you