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

COMP5703 Seminar Approximate Max-Flow Min- Cut Theorems and Applications Shanshan Wang Nov.17 th , 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]. Source from [5 ] • 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 𝑡 𝑗 , sink 𝑢 𝑗 and demand 𝐸 𝑗 . Source from [5 ]

Introduction Objective : Simultaneously route 𝐸 𝑗 units of commodity 𝑗 from 𝑡 𝑗 to 𝑢 𝑗 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 𝑔𝐸 𝑗 units 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: 𝐷(𝑉, 𝑉) 𝜔 = 𝑛𝑗𝑜 𝑉⊆𝑊 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 • 𝝎 and min-cut 𝝎 for which 𝒈 ≤ 𝑷 𝒎𝒑𝒉 𝒐 . • Illustration by example: Let G be a 3-regular 𝑜 -node graph with unit edge capacities for which > | ≥ 𝑑 min{ 𝑉 , |𝑉 |} | < 𝑉, 𝑉 for some constant 𝑑 > 0 and all 𝑉 ⊆ 𝑊 . Thus - Min-cut 𝜔 : > | 𝐷(𝑉, 𝑉) | < 𝑉, 𝑉 𝜔 = 𝑛𝑗𝑜 𝑉⊆𝑊 = 𝑛𝑗𝑜 𝑉⊆𝑊 | |𝑉||𝑉 𝐸(𝑉, 𝑉) 𝑑 𝑑 ≥ 𝑛𝑗𝑜 𝑉⊆𝑊 {|𝑉|,|𝑉|} = 𝑜−1 . max 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. 1 2 𝑜 2 𝑔(log 𝑜 − 2). (4) Thus, the capacity in the network must be at least (5) Since the graph is 3-regular and has unit capacity edges, the total 3𝑜 6 capacity is at most 3𝑜/2 , thus 𝑔 ≤ 2 𝑔(log 𝑜−2) = (𝑜−1)(log 𝑜−2) . 𝑜 𝑑 (6) According to min-cut, 𝑜 − 1 ≥ 𝜔 , then 6 𝑑(log 𝑜−2) = 𝑃( 𝜔 6𝜔 𝑔 ≤ (𝑜−1)(log 𝑜−2) ≤ log 𝑜 ) .

Approximate Max-Flow Min-Cut • Theorem 2 : For any UMFP, 𝝎 𝛁 ≤ 𝒈 ≤ 𝝎 𝒎𝒑𝒉 𝒐 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) denote total 𝑓∈𝐹 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 ′ to be the subgraph of 𝐻′ consisting of nodes and edges within distance 𝑗 𝐻 𝑗 2𝐷 of 𝑤 . Let 𝐷 0 = 𝑜 , and for 𝑗 > 0 , define 𝐷 𝑗 to be the total capacity of the ′ . Let 𝑘 denote the smallest value of 𝑗 ≥ 0 for which 𝐷 𝑗+1 < (1 + edges in 𝐻 𝑗 ′ form the 𝜗) 𝐷 𝑗 where 𝜗 = (𝑋𝑚𝑝𝑕 𝑜/𝛦𝐷 )<1/4. Now the nodes and edges in 𝐻 𝑘 first component. ′ from 𝐻′ and repeat (2) until there are no longer any nodes 𝑤 (3) Remove 𝐻 𝑘 of 𝐻′ that correspond to nodes in 𝐻. Let 𝐷′ denote the total initial capacity of 𝐻 ′ , then 𝐷 ′ = 𝐷𝑒 𝑓 𝐷 𝑓 + 𝐷 𝑋 𝐷 𝑓 ≤ 𝐷 𝑓 𝑒(𝑓) = 2𝐷. 𝑓∈𝐹 𝑓∈𝐹 𝑓∈𝐹 𝑋

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 ′ + 𝑜𝐷 0 ≤ 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 = Δ 𝐷𝜗 ∎

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 1/2𝑜 2 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 = 1/2𝑜 2 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 = 1/2𝑜 2 . (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 1/2𝑜 2 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 >| > for which |<𝑉,𝑉 The sparsest cut of a graph 𝐻 = (𝑊, 𝐹) is a partition < 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: ) 𝐷(𝑉,𝑉 𝛽 = 𝑛𝑗𝑜 𝑉⊆𝑊 ) . 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……