Approximate Max-Flow Min- Cut Theorems and Applications Shanshan - - PowerPoint PPT Presentation

Approximate Max-Flow Min- Cut Theorems and Applications

COMP5703 Seminar Shanshan Wang Nov.17th, 2014

Contents

• Introduction

– Single and Multicommodity Flow Problem – Uniform Multicommodity Flow Problem

• Approximate Max-Flow Min-Cut

– Two Theorems for Uniform Multicommodity Flow Problem and Their Proofs

• Applications

– Sparsest Cut – Flux, Minimum Quotient Separators

• References

Source from: http://newsoffice.mit.edu/2013/new- algorithm-can-dramatically-streamline-solutions-to-the- max-flow-problem-0107

Introduction

• Single Commodity Flow Problem [Ford and Fulkerson 1956].
• Multicommodity Flow Problem (MFP) is a network problem with

multiple commodities between different source and sink nodes. There are 𝑙 ≥ 1 commodities, each with source 𝑡𝑗, sink 𝑢𝑗 and demand 𝐸𝑗.

Source from [5] Source from [5]

Introduction

Objective: Simultaneously route 𝐸𝑗 units of commodity 𝑗 from 𝑡𝑗 to 𝑢𝑗 for each 𝑗 so that the total amount of all commodities passing through any edge is no greater than its capacity. Max-flow is the maximum value of 𝑔 such that 𝑔𝐸𝑗units of commodity 𝑗 can be simultaneously routed for each 𝑗 without violating any capacity constraints, where 𝑔 is the common fraction of each commodity that is routed.

Min-cut is the minimum of all cuts of the ratio of the capacity of the cut to the demand of the cut, denoted as: 𝜔 = 𝑛𝑗𝑜𝑉⊆𝑊 𝐷(𝑉, 𝑉) 𝐸(𝑉, 𝑉) where 𝐷 𝑉, 𝑉 is the sum of capacities of the edges linking 𝑉 to 𝑉, and 𝐸 𝑉, 𝑉 is the sum of the demands whose source and sink are on opposite sides of the cut that separates 𝑉 from 𝑉. * Note: Max-flow is always upper bounded by the min-cut for MFP.

Introduction

• In Uniform MFP (UMFP), there is a commodity for every pair of nodes

and the demand for every commodity is the same (usually set to 1).

• Approximate Max-Flow and Min-Cut Theorems (Tom Leighton and Satish

Rao 1999, 1988 [5][4])

• Theorem 1: For UMFP, the max-flow is a Ω(log 𝑜)-factor smaller than

the min-cut.

• Theorem 2 (A tight bound): For UMFP, the max-flow is always within

a Θ(log 𝑜)-factor of the min-cut.

Approximate Max-Flow Min-Cut

• Theorem 1: For any 𝒐, there is an 𝒐-node UMFP with max-flow 𝒈

and min-cut 𝝎 for which 𝒈 ≤ 𝑷

𝝎 𝒎𝒑𝒉 𝒐 .

• Illustration by example:

Let G be a 3-regular 𝑜-node graph with unit edge capacities for which | < 𝑉, 𝑉 > | ≥ 𝑑 min{ 𝑉 , |𝑉 |} for some constant 𝑑 > 0 and all 𝑉 ⊆ 𝑊. Thus

• Min-cut 𝜔:

𝜔 = 𝑛𝑗𝑜𝑉⊆𝑊 𝐷(𝑉, 𝑉) 𝐸(𝑉, 𝑉) = 𝑛𝑗𝑜𝑉⊆𝑊 | < 𝑉, 𝑉 > | |𝑉||𝑉 | ≥ 𝑛𝑗𝑜𝑉⊆𝑊

𝑑 max {|𝑉|,|𝑉|} = 𝑑 𝑜−1.

i.e. 𝜔 ≥ 𝑑

𝑜−1

Approximate Max-Flow Min-Cut

• Max-flow 𝑔:

(1) Since 𝐻 is 3-regular, there are at most 𝑜/2 nodes within distance log 𝑜 − 3 of any particular node 𝑤 ∈ 𝑊. (2) Hence, for at least half of the 𝑜

2 commodities, the shortest path

connecting the source and sink in 𝐻 has at least log 𝑜 − 2 edges. (3) In order to sustain a flow of 𝑔 for such a commodity, at least 𝑔(log 𝑜 − 2) capacity must be used by the commodity. (4) Thus, the capacity in the network must be at least

1 2 𝑜 2 𝑔(log 𝑜 − 2).

(5) Since the graph is 3-regular and has unit capacity edges, the total capacity is at most 3𝑜/2, thus 𝑔 ≤

3𝑜

𝑜 2 𝑔(log 𝑜−2) =

6 (𝑜−1)(log 𝑜−2).

(6) According to min-cut, 𝑜 − 1 ≥

𝑑 𝜔, then

𝑔 ≤

6 (𝑜−1)(log 𝑜−2) ≤ 6𝜔 𝑑(log 𝑜−2) = 𝑃( 𝜔 log 𝑜).

Approximate Max-Flow Min-Cut

• Theorem 2: For any UMFP,

𝛁 𝝎 𝒎𝒑𝒉 𝒐 ≤ 𝒈 ≤ 𝝎 where 𝒈 is the max-flow and 𝝎 is the min-cut of the UMFP.

• General approach:

(1) Max-Flow: from the duality theory of LP, an optimal distance function results in a total weight that is equal to the max-flow of the UMFP. (2) Min-Cut: 3-stage process: – Stage 1: Consider the dual of UMFP and use the optimal solution to define a graph with distance labels on the edges. – Stage 2: Starting from a source or a sink, grow a region in the graph until find a cut of small enough capacity separating the root from its mate. – Stage 3: The region is removed and the process is repeated.

Approximate Max-Flow Min-Cut

• Lemma 3. For any graph 𝐻 with arbitrary edge capacities, any 𝛦 > 0,

and any distance function with total weight 𝑋, it is possible to partition 𝐻 into components with radius at most 𝛦 so that the capacity of the edges connecting nodes in different components is at most 4𝑋𝑚𝑝𝑕 𝑜/ 𝛦.

• Proof:

Let 𝐷 = 𝐷(𝑓)

𝑓∈𝐹

denote total capacity on the edges of 𝐻.

• If 𝛦 ≤ 4𝑋𝑚𝑝𝑕 𝑜/𝐷, done.
• If 𝛦 > 4𝑋𝑚𝑝𝑕 𝑜/𝐷, then

(1) construct 𝐻′ from 𝐻 by replacing each edge 𝑓 of 𝐻 with a path of 𝐷𝑒(𝑓)/𝑋 edges.

Approximate Max-Flow Min-Cut

(2) Form components of 𝐻′: select any node 𝑤 in 𝐻′ that corresponds to a node in 𝐻. For each 𝑗 ≥ 0, let 𝐻𝑗

′ to be the subgraph of 𝐻′ consisting of nodes and edges within distance 𝑗

• f 𝑤. Let 𝐷0 =

2𝐷 𝑜 , and for 𝑗 > 0, define 𝐷𝑗 to be the total capacity of the

edges in 𝐻𝑗

′. Let 𝑘 denote the smallest value of 𝑗 ≥ 0 for which 𝐷𝑗+1 < (1 +

𝜗) 𝐷𝑗 where 𝜗 = (𝑋𝑚𝑝𝑕 𝑜/𝛦𝐷)<1/4. Now the nodes and edges in 𝐻

𝑘 ′ form the

first component. (3) Remove 𝐻

𝑘 ′ from 𝐻′ and repeat (2) until there are no longer any nodes 𝑤

• f 𝐻′ that correspond to nodes in 𝐻.

Let 𝐷′ denote the total initial capacity of 𝐻′, then 𝐷′ = 𝐷 𝑓

𝐷𝑒 𝑓 𝑋

≤

𝑓∈𝐹

𝐷 𝑓 + 𝐷

𝑋

𝐷 𝑓 𝑒(𝑓)

𝑓∈𝐹

= 2𝐷.

𝑓∈𝐹

Approximate Max-Flow Min-Cut

Also, we know capacity of the edges leaving any component ≤ 𝜗𝐷

𝑘.

Thus total capacity on all edges leaving all components in 𝐻′ is at most 𝜗 𝐷′ + 𝑜𝐷0 ≤ 𝜗 2𝐷 + 2𝐷 = 4𝜗𝐷. Any edge of 𝐻′ that links two components of 𝐻 must correspond to a path of capacity 𝐷 𝑓 edges in 𝐻′ that was cut to form at least one component of 𝐻’. Hence, the total capacity of the edges linking different components in 𝐻 is at most 4𝜗𝐷 = (4𝑋𝑚𝑝𝑕 𝑜/𝛦). Thus the radius of each component in 𝐻 is at most 𝑋𝑚𝑝𝑕 𝑜 𝐷𝜗 = Δ ∎

Approximate Max-Flow Min-Cut

Ratio cost of a cut < 𝑉, 𝑉 > is the quantity 𝐷(𝑉,𝑉)

|𝑉||𝑉 | .

Corollary 4. For any graph 𝐻 and any distance function with total weight 𝑋, we can either (1) find a component with radius 1/2𝑜2 that contains at least 2/3 of the nodes in 𝐻 or (2) find a cut of 𝐻 with ratio cost 𝑃 𝑋𝑚𝑝𝑕 𝑜 . Proof: Apply the result of Lemma 3 with Δ = 1/2𝑜2 then discuss both cases. Lemma 5. For any graph 𝐻, if there is a distance function 𝑒 with total weight 𝑋 and a subset of nodes 𝑈 ⊆ 𝑊 with 𝑈 ≥ 2𝑜/3 and 𝑒 𝑈, 𝑣 ≥ 1/2𝑜

𝑣∈𝑊−𝑈

Then we can find a cut with ratio cost 𝑃 𝑋 . Proof: Same idea of subgraph construction of Lemma 3.

Approximate Max-Flow Min-Cut

Lemma 6. For any graph 𝐻 with total weight 𝑋 that satisfies the distance constraint, we can find a cut with ratio cost 𝑃 𝑋𝑚𝑝𝑕 𝑜 . Proof: (1) First partition graph 𝐻 as in Lemma 3 with Δ = 1/2𝑜2. (2) By Corollary 4, we can then either find a cut with ratio cost 𝑃 𝑋𝑚𝑝𝑕 𝑜 (then we are done) or find a component 𝑈 with radius 1/2𝑜2 that contains at least 2𝑜/3 nodes. (3) If it is the latter case of (2), then apply Lemma 5 to find a cut with ratio cost 𝑃 𝑋 . ∎ Note:

• The proof of Theorem 2 follows from Lemma 6 and the fact that 𝑋 = 𝑔.
• All the proofs can be transformed to polynomial time algorithms for finding

cuts.

Applications

The sparsest cut of a graph 𝐻 = (𝑊, 𝐹) is a partition < 𝑉, 𝑉 > for which |<𝑉,𝑉

>| |𝑉||𝑉 |

is minimized.

• NP-hard.
• Can be approximated to within an 𝑃(log 𝑜) factor.
• By setting all demands and capacities to be 1 and find a cut with ratio cost

𝑃(𝑔𝑚𝑝𝑕 𝑜). Flux or minimum edge expansion of a graph is defined by: 𝛽 = 𝑛𝑗𝑜𝑉⊆𝑊

𝐷(𝑉,𝑉 ) min ( 𝑉 , 𝑉 ) .

• A graph has flux at least 𝛽 if every subset 𝑉 with at most half of the nodes

is connected to the rest of the graph with edges of total weight at least 𝛽|𝑉|.

• A cut that achieves the flux is minimum quotient separator (NP-hard).
• Can be approximated to within an 𝑃(log 𝑜) factor.

Many more……

References

1. Sanjeev Khanna Chandra Chekuri and F.Bruce Shepherd. The all-or-nothing multicommodity flow problem. Proc. of ACM STOC,2004. 2. M.R. Garey and D.S. Johnson. Computers and intractability: A guide to the theory of np-

• completeness. Freeman, San Francisco, Calif, 1979.

3. Jon Kleinberg and Ronitt Rubinfeld. Short paths in expander graphs. 1996 IEEE Symposium on Foundations of Computer Science, 1996. 4. Tom Leighton and Satish Rao. An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms. in Proc.29th IEEE Symposium on Foundations of Computer Science, pages 422-431, 1988. 5. Tom Leighton and Satish Rao. Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. J.ACM, 46:787-832, 1999. 6.

• F. Matula, D.W.and Shahrokhi. The maximum concurrent flow problem and sparsest cuts.

Tech.Rep.Southern Methodist Univ.,Dallas,Tex, 1986. 7. Vijay V.Vazirani Naveen Garg and Mihalis Yannakakis. Approximate max-flow min-(multi)cut theorems and their applications. SIAM J. COMPUT., 25:235-251,1996. 8. R.Ravi Philip Klein, Ajit Agrawal and Satish Rao. Approximation through multicommodity flow. in Proc.31st IEEE Symposium on Foundations of Computer Science, pages 726-737, 1990. 9. R.Ravi Philip Klein, Ajit Agrawal and Satish Rao. Approximation through multi-commodity flow. Combinatorica, 15:187-202, 1995.

• 10. S.Tragoudas. Improved approximations for the minimum-cut ratio and the flux. Math.Systems

Theory, 29:157-167, 1996.