[PPT] - Delay-Constrained Unicast: Improved upper bounds Sudeep Kamath PowerPoint Presentation

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Delay-Constrained Unicast: Improved upper bounds

Sudeep Kamath

Joint work with Chandra Chekuri Sreeram Kannan Pramod Viswanath

DIMACS workshop on Network Coding, 17 December 2015

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Delay-constrained unicast [Wang-Chen '14]

Practical constraint - eg. video streaming, fjnancial data Intra-fmow coding has fewer security and privacy concerns Implementation aligned with self-interest

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Single flow with delay constraint D Delay-constrained unicast [Wang-Chen '14]

Practical constraint - eg. video streaming, fjnancial data Intra-fmow coding has fewer security and privacy concerns Implementation aligned with self-interest

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Single flow with delay constraint D Delay-constrained unicast [Wang-Chen '14]

Practical constraint - eg. video streaming, fjnancial data Intra-fmow coding has fewer security and privacy concerns Implementation aligned with self-interest

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Single flow with delay constraint D For this network with D=6, Capacity Flow = 4 3 Delay-constrained unicast [Wang-Chen '14]

Practical constraint - eg. video streaming, fjnancial data Intra-fmow coding has fewer security and privacy concerns Implementation aligned with self-interest

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Single flow with delay constraint D For this network with D=6, Capacity Flow = 4 3 Delay-constrained unicast [Wang-Chen '14]

Practical constraint - eg. video streaming, fjnancial data Intra-fmow coding has fewer security and privacy concerns Implementation aligned with self-interest

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Single flow with delay constraint D For this network with D=6, Capacity Flow = 4 3 Delay-constrained unicast [Wang-Chen '14]

Practical constraint - eg. video streaming, fjnancial data Intra-fmow coding has fewer security and privacy concerns Implementation aligned with self-interest

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Single flow with delay constraint D For this network with D=6, Capacity Flow = 4 3 Delay-constrained unicast [Wang-Chen '14]

Practical constraint - eg. video streaming, fjnancial data Intra-fmow coding has fewer security and privacy concerns Implementation aligned with self-interest

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Single flow with delay constraint D For this network with D=6, Capacity Flow = 4 3 Delay-constrained unicast [Wang-Chen '14] Practical constraint - eg. video streaming, fjnancial data Intra-fmow coding has fewer security and privacy concerns Implementation aligned with self-interest

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Single flow with delay constraint D For this network with D=6, Capacity Flow = 4 3 Delay-constrained unicast [Wang-Chen '14] How large can this ratio be? Practical constraint - eg. video streaming, fjnancial data Intra-fmow coding has fewer security and privacy concerns Implementation aligned with self-interest

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Single flow with delay constraint D For this network with D=6, Capacity Flow = 4 3 Delay-constrained unicast [Wang-Chen '14] How large can this ratio be? 2 ≤ sup

graphs

Capacity Flow ≤ D + 1 Practical constraint - eg. video streaming, fjnancial data Intra-fmow coding has fewer security and privacy concerns Implementation aligned with self-interest

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Single flow with delay constraint D For this network with D=6, Capacity Flow = 4 3 Delay-constrained unicast [Wang-Chen '14] How large can this ratio be? 2 ≤ sup

graphs

Capacity Flow ≤ D + 1 We improve

ver this

Practical constraint - eg. video streaming, fjnancial data Intra-fmow coding has fewer security and privacy concerns Implementation aligned with self-interest

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Combinatorial Optimization Network Information Theory This work

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Combinatorial Optimization Network Information Theory This work

Multi-commodity flow problem

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Combinatorial Optimization Network Information Theory This work

Multi-commodity flow problem Multiple-unicast problem

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Multi-commodity fmow / Multiple-unicast

Given a directed graph and k source-destination pairs {(si, di)} Flow: Maximum total commodity fmow EdgeCut: Fewest edges whose removal disconnects all paths from to Capacity: Maximum information fmow EdgeCut Cutset bound Capacity Cutset bound However, we may have EdgeCut Capacity

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Multi-commodity fmow / Multiple-unicast

Given a directed graph and k source-destination pairs {(si, di)} Flow: Maximum total commodity fmow EdgeCut: Fewest edges whose removal disconnects all paths from to Capacity: Maximum information fmow EdgeCut Cutset bound Capacity Cutset bound However, we may have EdgeCut Capacity s1 s2

d2 d1

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Multi-commodity fmow / Multiple-unicast

Given a directed graph and k source-destination pairs {(si, di)} Flow: Maximum total commodity fmow EdgeCut: Fewest edges whose removal disconnects all paths from to Capacity: Maximum information fmow EdgeCut Cutset bound Capacity Cutset bound However, we may have EdgeCut Capacity s1 s2

d2 d1

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Multi-commodity fmow / Multiple-unicast

Given a directed graph and k source-destination pairs {(si, di)} Flow: Maximum total commodity fmow EdgeCut: Fewest edges whose removal disconnects all paths from to Capacity: Maximum information fmow EdgeCut Cutset bound Capacity Cutset bound However, we may have EdgeCut Capacity s1 s2

d2 d1 Flow = 1

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Multi-commodity fmow / Multiple-unicast

Given a directed graph and k source-destination pairs {(si, di)} Flow: Maximum total commodity fmow EdgeCut: Fewest edges whose removal disconnects all paths from si to di ∀i Capacity: Maximum information fmow EdgeCut Cutset bound Capacity Cutset bound However, we may have EdgeCut Capacity s1 s2

d2 d1 Flow = 1

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Multi-commodity fmow / Multiple-unicast

Given a directed graph and k source-destination pairs {(si, di)} Flow: Maximum total commodity fmow EdgeCut: Fewest edges whose removal disconnects all paths from si to di ∀i Capacity: Maximum information fmow EdgeCut Cutset bound Capacity Cutset bound However, we may have EdgeCut Capacity s1 s2

×

d2 d1 Flow = 1

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Multi-commodity fmow / Multiple-unicast

Given a directed graph and k source-destination pairs {(si, di)} Flow: Maximum total commodity fmow EdgeCut: Fewest edges whose removal disconnects all paths from si to di ∀i Capacity: Maximum information fmow EdgeCut Cutset bound Capacity Cutset bound However, we may have EdgeCut Capacity s1 s2

× × ×

d2 d1 Flow = 1 EdgeCut = 1

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Multi-commodity fmow / Multiple-unicast

Given a directed graph and k source-destination pairs {(si, di)} Flow: Maximum total commodity fmow EdgeCut: Fewest edges whose removal disconnects all paths from si to di ∀i Capacity: Maximum information fmow EdgeCut Cutset bound Capacity Cutset bound However, we may have EdgeCut Capacity s1 s2

× × ×

d2 d1 Flow = 1 EdgeCut = 1

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Multi-commodity fmow / Multiple-unicast

Given a directed graph and k source-destination pairs {(si, di)} Flow: Maximum total commodity fmow EdgeCut: Fewest edges whose removal disconnects all paths from si to di ∀i Capacity: Maximum information fmow EdgeCut Cutset bound Capacity Cutset bound However, we may have EdgeCut Capacity s1 s2

d2 d1 Flow = 1 EdgeCut = 1 a ⊕ b a b a ⊕ b a ⊕ b a b

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Multi-commodity fmow / Multiple-unicast

Given a directed graph and k source-destination pairs {(si, di)} Flow: Maximum total commodity fmow EdgeCut: Fewest edges whose removal disconnects all paths from si to di ∀i Capacity: Maximum information fmow EdgeCut Cutset bound Capacity Cutset bound However, we may have EdgeCut Capacity s1 s2

d2 d1 Flow = 1 EdgeCut = 1 a ⊕ b a b a ⊕ b a ⊕ b a b Capacity = 2

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Multi-commodity fmow / Multiple-unicast

Given a directed graph and k source-destination pairs {(si, di)} Flow: Maximum total commodity fmow EdgeCut: Fewest edges whose removal disconnects all paths from si to di ∀i Capacity: Maximum information fmow EdgeCut ̸= Cutset bound Capacity Cutset bound However, we may have EdgeCut Capacity s1 s2

d2 d1 Flow = 1 EdgeCut = 1 a ⊕ b a b a ⊕ b a ⊕ b a b Capacity = 2

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Multi-commodity fmow / Multiple-unicast

Given a directed graph and k source-destination pairs {(si, di)} Flow: Maximum total commodity fmow EdgeCut: Fewest edges whose removal disconnects all paths from si to di ∀i Capacity: Maximum information fmow EdgeCut ̸= Cutset bound Capacity ≤ Cutset bound However, we may have EdgeCut Capacity s1 s2

d2 d1 Flow = 1 EdgeCut = 1 a ⊕ b a b a ⊕ b a ⊕ b a b Capacity = 2

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Multi-commodity fmow / Multiple-unicast

Given a directed graph and k source-destination pairs {(si, di)} Flow: Maximum total commodity fmow EdgeCut: Fewest edges whose removal disconnects all paths from si to di ∀i Capacity: Maximum information fmow EdgeCut ̸= Cutset bound Capacity ≤ Cutset bound However, we may have EdgeCut < Capacity s1 s2

d2 d1 Flow = 1 EdgeCut = 1 a ⊕ b a b a ⊕ b a ⊕ b a b Capacity = 2

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Flow = EdgeCut = Capacity

For k = 1:

(Max-Flow Min-Cut Theorem)

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Flow = EdgeCut = Capacity

For k = 1:

(Max-Flow Min-Cut Theorem)

For k ≥ 2:

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Flow ≤ EdgeCut Flow ≤ Capacity EdgeCut ≶ Capacity Flow = EdgeCut = Capacity

For k = 1:

(Max-Flow Min-Cut Theorem)

For k ≥ 2:

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EdgeCut Capacity Flow Flow ≤ EdgeCut Flow ≤ Capacity EdgeCut ≶ Capacity Flow = EdgeCut = Capacity

For k = 1:

(Max-Flow Min-Cut Theorem)

For k ≥ 2:

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EdgeCut Capacity Flow

[Saks-Samorodnitsky-Zosin '04], [Harvey-Kleinberg-Lehman '06], [Ambühl-Mastrolilli-Svensson '07], [Chuzhoy-Khanna '07], [Dougherty-Freiling-Zeger '05], [Chan-Grant '08]

Flow ≤ EdgeCut Flow ≤ Capacity EdgeCut ≶ Capacity Flow = EdgeCut = Capacity

For k = 1:

(Max-Flow Min-Cut Theorem)

For k ≥ 2:

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EdgeCut Capacity Flow

[Saks-Samorodnitsky-Zosin '04], [Harvey-Kleinberg-Lehman '06], [Ambühl-Mastrolilli-Svensson '07], [Chuzhoy-Khanna '07], [Dougherty-Freiling-Zeger '05], [Chan-Grant '08]

Linear Program

Flow ≤ EdgeCut Flow ≤ Capacity EdgeCut ≶ Capacity Flow = EdgeCut = Capacity

For k = 1:

(Max-Flow Min-Cut Theorem)

For k ≥ 2:

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EdgeCut Capacity Flow

[Saks-Samorodnitsky-Zosin '04], [Harvey-Kleinberg-Lehman '06], [Ambühl-Mastrolilli-Svensson '07], [Chuzhoy-Khanna '07], [Dougherty-Freiling-Zeger '05], [Chan-Grant '08]

Linear Program NP-hard, hard to approximate

Flow ≤ EdgeCut Flow ≤ Capacity EdgeCut ≶ Capacity Flow = EdgeCut = Capacity

For k = 1:

(Max-Flow Min-Cut Theorem)

For k ≥ 2:

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EdgeCut Capacity Flow

[Saks-Samorodnitsky-Zosin '04], [Harvey-Kleinberg-Lehman '06], [Ambühl-Mastrolilli-Svensson '07], [Chuzhoy-Khanna '07], [Dougherty-Freiling-Zeger '05], [Chan-Grant '08]

Linear Program NP-hard, hard to approximate

Flow ≤ EdgeCut Flow ≤ Capacity EdgeCut ≶ Capacity Flow = EdgeCut = Capacity

For k = 1:

(Max-Flow Min-Cut Theorem)

For k ≥ 2:

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EdgeCut Capacity Flow

[Saks-Samorodnitsky-Zosin '04], [Harvey-Kleinberg-Lehman '06], [Ambühl-Mastrolilli-Svensson '07], [Chuzhoy-Khanna '07], [Dougherty-Freiling-Zeger '05], [Chan-Grant '08]

Linear Program NP-hard, hard to approximate

Flow ≤ EdgeCut Flow ≤ Capacity EdgeCut ≶ Capacity Flow = EdgeCut = Capacity

For k = 1:

(Max-Flow Min-Cut Theorem)

For k ≥ 2:

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EdgeCut Capacity Flow

[Saks-Samorodnitsky-Zosin '04], [Harvey-Kleinberg-Lehman '06], [Ambühl-Mastrolilli-Svensson '07], [Chuzhoy-Khanna '07], [Dougherty-Freiling-Zeger '05], [Chan-Grant '08]

Linear Program NP-hard, hard to approximate

Flow ≤ EdgeCut Flow ≤ Capacity EdgeCut ≶ Capacity Flow = EdgeCut = Capacity

For k = 1:

(Max-Flow Min-Cut Theorem)

For k ≥ 2:

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EdgeCut Capacity Flow

[Saks-Samorodnitsky-Zosin '04], [Harvey-Kleinberg-Lehman '06], [Ambühl-Mastrolilli-Svensson '07], [Chuzhoy-Khanna '07], [Dougherty-Freiling-Zeger '05], [Chan-Grant '08]

Linear Program NP-hard, hard to approximate

Linear coding not sufficient, entropic cone necessary

Flow ≤ EdgeCut Flow ≤ Capacity EdgeCut ≶ Capacity Flow = EdgeCut = Capacity

For k = 1:

(Max-Flow Min-Cut Theorem)

For k ≥ 2:

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EdgeCut Capacity Flow Can be multiplicative factor k apart

[Saks-Samorodnitsky-Zosin '04], [Harvey-Kleinberg-Lehman '06], [Ambühl-Mastrolilli-Svensson '07], [Chuzhoy-Khanna '07], [Dougherty-Freiling-Zeger '05], [Chan-Grant '08]

Linear Program NP-hard, hard to approximate

Linear coding not sufficient, entropic cone necessary

Flow ≤ EdgeCut Flow ≤ Capacity EdgeCut ≶ Capacity Flow = EdgeCut = Capacity

For k = 1:

(Max-Flow Min-Cut Theorem)

For k ≥ 2:

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Triangle-cast

Main Result 1: For triangle-cast as above,

EdgeCut log Flow Capacity EdgeCut

Main ideas: Adaptation of a “region-growing” technique [Garg-Vazirani-Yannakakis ’96] Generalized Network Sharing bound [K.-Tse-Anantharam ’11]

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Triangle-cast

s1 s2 sk d1 d2 dk Directed Graph Multiple-unicast: flow from si to di for all i

Main Result 1: For triangle-cast as above,

EdgeCut log Flow Capacity EdgeCut

Main ideas: Adaptation of a “region-growing” technique [Garg-Vazirani-Yannakakis ’96] Generalized Network Sharing bound [K.-Tse-Anantharam ’11]

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Triangle-cast

s1 s2 sk d1 d2 dk Directed Graph Multiple-unicast: flow from si to di for all i Triangle-cast: flow from si to dj for all i ≥ j

Main Result 1: For triangle-cast as above,

EdgeCut log Flow Capacity EdgeCut

Main ideas: Adaptation of a “region-growing” technique [Garg-Vazirani-Yannakakis ’96] Generalized Network Sharing bound [K.-Tse-Anantharam ’11]

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Triangle-cast

s1 s2 sk d1 d2 dk Directed Graph Multiple-unicast: flow from k(k + 1) 2 flows si to di for all i Triangle-cast: flow from si to dj for all i ≥ j

Main Result 1: For triangle-cast as above,

EdgeCut log Flow Capacity EdgeCut

Main ideas: Adaptation of a “region-growing” technique [Garg-Vazirani-Yannakakis ’96] Generalized Network Sharing bound [K.-Tse-Anantharam ’11]

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Triangle-cast

s1 s2 sk d1 d2 dk Directed Graph Multiple-unicast: flow from k(k + 1) 2 flows si to di for all i Triangle-cast: flow from si to dj for all i ≥ j

Main Result 1: For triangle-cast as above,

EdgeCut 4 loge(k + 1) ≤ Flow ≤ Capacity ≤ EdgeCut

Main ideas: Adaptation of a “region-growing” technique [Garg-Vazirani-Yannakakis ’96] Generalized Network Sharing bound [K.-Tse-Anantharam ’11]

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Triangle-cast

s1 s2 sk d1 d2 dk Directed Graph Multiple-unicast: flow from k(k + 1) 2 flows si to di for all i Triangle-cast: flow from si to dj for all i ≥ j

Main Result 1: For triangle-cast as above,

EdgeCut 4 loge(k + 1) ≤ Flow ≤ Capacity ≤ EdgeCut

Main ideas: Adaptation of a “region-growing” technique [Garg-Vazirani-Yannakakis ’96] Generalized Network Sharing bound [K.-Tse-Anantharam ’11]

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Triangle-cast

s1 s2 sk d1 d2 dk Directed Graph Multiple-unicast: flow from k(k + 1) 2 flows si to di for all i Triangle-cast: flow from si to dj for all i ≥ j

Main Result 1: For triangle-cast as above,

EdgeCut 4 loge(k + 1) ≤ Flow ≤ Capacity ≤ EdgeCut

Main ideas: Adaptation of a “region-growing” technique [Garg-Vazirani-Yannakakis ’96] Generalized Network Sharing bound [K.-Tse-Anantharam ’11]

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s

d Delay constraint D = 2

Delay constrained network

Main Result 2 Capacity Flow log

Delay

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Reproduced from xkcd.com

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s

d Delay constraint D = 2

Delay constrained network

Main Result 2 Capacity Flow log

Delay

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s(1) d(1) d(2) d(3) d(4) d(5) s(2) s(3) s(4) s(5)

s1 s2 d2 d1 s3 d3

Time 0 Time 1 Time 2 Time D Time D+1 Time D+2

s

d Delay constraint D = 2

Delay constrained network

Main Result 2 Capacity Flow log

Delay

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s(1) d(1) d(2) d(3) d(4) d(5) s(2) s(3) s(4) s(5)

s1 s2 d2 d1 s3 d3

Time 0 Time 1 Time 2 Time D Time D+1 Time D+2

s

d Delay constraint D = 2

Delay constrained network

Main Result 2 Capacity Flow log

Delay

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s(1) d(1) d(2) d(3) d(4) d(5) s(2) s(3) s(4) s(5)

s1 s2 d2 d1 s3 d3

Time 0 Time 1 Time 2 Time D Time D+1 Time D+2

Multiple-unicast s

d Delay constraint D = 2

Delay constrained network

Main Result 2 Capacity Flow log

Delay

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s(1) d(1) d(2) d(3) d(4) d(5) s(2) s(3) s(4) s(5)

s1 s2 d2 d1 s3 d3

Time 0 Time 1 Time 2 Time D Time D+1 Time D+2

Multiple-unicast Information delivered earlier is ok! s

d Delay constraint D = 2

Delay constrained network

Main Result 2 Capacity Flow log

Delay

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s(1) d(1) d(2) d(3) d(4) d(5) s(2) s(3) s(4) s(5)

s1 s2 d2 d1 s3 d3

Time 0 Time 1 Time 2 Time D Time D+1 Time D+2

Multiple-unicast Information delivered earlier is ok! s

d Delay constraint D = 2

Delay constrained network

Main Result 2 Capacity Flow log

Delay

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s(1) d(1) d(2) d(3) d(4) d(5) s(2) s(3) s(4) s(5)

s1 s2 d2 d1 s3 d3

Time 0 Time 1 Time 2 Time D Time D+1 Time D+2

Multiple-unicast Information delivered earlier is ok! Hence, triangle-cast! s

d Delay constraint D = 2

Delay constrained network

Main Result 2 Capacity Flow log

Delay

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s(1) d(1) d(2) d(3) d(4) d(5) s(2) s(3) s(4) s(5)

s1 s2 d2 d1 s3 d3

Time 0 Time 1 Time 2 Time D Time D+1 Time D+2

Multiple-unicast Information delivered earlier is ok! Hence, triangle-cast! s

d Delay constraint D = 2

Delay constrained network

Main Result 2 Capacity Flow ≤ 8 loge(D+1)

D = Delay

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Open 1: “Multicast”

s d1 d2 d3

Multicast: same information to all destinations What happens with delay constraint? Practical constraint Intra-fmow coding Coding strategies? - Random coding does not work

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Open 1: “Multicast”

s d1 d2 d3

Multicast: same information to all destinations What happens with delay constraint? Practical constraint Intra-fmow coding Coding strategies? - Random coding does not work

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Open 1: “Multicast”

s d1 d2 d3

Multicast: same information to all destinations What happens with delay constraint? Practical constraint Intra-fmow coding Coding strategies? - Random coding does not work

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Open 2: “Triangle-cast gap”

Theorem (this work)

For k-triangle-cast, EdgeCut 4 loge(k + 1) ≤ Flow ≤ Capacity ≤ EdgeCut

Conjecture

For

triangle-cast,

EdgeCut Flow Capacity EdgeCut Hence, for delay-constrained unicast, Capacity Flow

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Open 2: “Triangle-cast gap”

Theorem (this work)

For k-triangle-cast, EdgeCut 4 loge(k + 1) ≤ Flow ≤ Capacity ≤ EdgeCut

Conjecture

For k-triangle-cast, EdgeCut 2 ≤ Flow ≤ Capacity ≤ EdgeCut Hence, for delay-constrained unicast, Capacity Flow ≤ 2

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Open 3: “Symmetry Principle”

Principle Under suitable symmetry in traffjc pattern, Flow, EdgeCut, Capacity are all not “too far” apart.

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Open 3: “Symmetry Principle”

F : Flow EC : EdgeCut C : Capacity

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Open 3: “Symmetry Principle”

F : Flow EC : EdgeCut C : Capacity

Bidirected Networks Symmetric Demands Group-cast Triangle-cast

[Leighton-Rao '88] [Linial-London-Rabinovich '94] [This work] [This work] [Klein-Plotkin-Rao-Tardos '93] [Naor-Zosin '01] [K.-Viswanath '12] [K.-Viswanath '12] [K.-Viswanath '12]

EC Θ(log k) ≤ F ≤ EC EC Θ(log3 k) ≤ F ≤ EC

EC 2 ≤ F ≤ EC

EC Θ(log k) ≤ F ≤ EC

F ≤ C ≤ EC F ≤ C ≤ 2 × EC F ≤ C ≤ EC F ≤ C ≤ EC

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Open 3: “Symmetry Principle”

F : Flow EC : EdgeCut C : Capacity

Bidirected Networks Symmetric Demands Group-cast Triangle-cast

[Leighton-Rao '88] [Linial-London-Rabinovich '94] [This work] [This work] [Klein-Plotkin-Rao-Tardos '93] [Naor-Zosin '01] [K.-Viswanath '12] [K.-Viswanath '12] [K.-Viswanath '12]

EC Θ(log k) ≤ F ≤ EC EC Θ(log3 k) ≤ F ≤ EC

EC 2 ≤ F ≤ EC

EC Θ(log k) ≤ F ≤ EC

F ≤ C ≤ EC F ≤ C ≤ 2 × EC F ≤ C ≤ EC F ≤ C ≤ EC

???

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Conclusion

s1 s2 sk d1 d2 dk Directed Graph Triangle-cast: flow from si to dj for all i ≥ j

= ⇒

Capacity Flow ≤ 8 loge(D + 1) Showed a 4 loge(k + 1) Flow − EdgeCut − Capacity approximation guarantee Delay-constrained unicast (delay D) has s1 s2 sk d1 d2 dk Directed Graph

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