AVERA RAGE GE SE SENSITIVIT SITIVITY OF OF GRA GRAPH PH ALGO - PowerPoint PPT Presentation

Related Sensitivity Notions ■ Diff ffer erent ential ial Priva vacy cy [Dwork McSherry Nissim Smith '06] – Edge Differential Privacy [Nissim Raskhodnikova Smith '07] An algorithm 𝐵 on a graph 𝐻 is differentially private if for all 𝑓 ∈ 𝐹 ■ the distributions 𝐵(𝐻) and 𝐵(𝐻 − 𝑓) are close to each other – Much stricter notion than average sensitivity – Some of our algorithms inspired by differentially private algorithms 33

Related Sensitivity Notions ■ Diff ffer erent ential ial Priva vacy cy [Dwork McSherry Nissim Smith '06] – Edge Differential Privacy [Nissim Raskhodnikova Smith '07] An algorithm 𝐵 on a graph 𝐻 is differentially private if for all 𝑓 ∈ 𝐹 ■ the distributions 𝐵(𝐻) and 𝐵(𝐻 − 𝑓) are close to each other – Much stricter notion than average sensitivity – Some of our algorithms inspired by differentially private algorithms ■ Stabi bilit lity y of f Learnin rning g Algo gorit rithms hms [Bousquet Elisseeff '02] 34

Related Sensitivity Notions ■ Diff ffer erent ential ial Priva vacy cy [Dwork McSherry Nissim Smith '06] – Edge Differential Privacy [Nissim Raskhodnikova Smith '07] An algorithm 𝐵 on a graph 𝐻 is differentially private if for all 𝑓 ∈ 𝐹 ■ the distributions 𝐵(𝐻) and 𝐵(𝐻 − 𝑓) are close to each other – Much stricter notion than average sensitivity – Some of our algorithms inspired by differentially private algorithms ■ Stabi bilit lity y of f Learnin rning g Algo gorit rithms hms [Bousquet Elisseeff '02] – A learner is stable if empirical loss does not change much by replacing any sample in the training data 35

Related Sensitivity Notions ■ Diff ffer erent ential ial Priva vacy cy [Dwork McSherry Nissim Smith '06] – Edge Differential Privacy [Nissim Raskhodnikova Smith '07] An algorithm 𝐵 on a graph 𝐻 is differentially private if for all 𝑓 ∈ 𝐹 ■ the distributions 𝐵(𝐻) and 𝐵(𝐻 − 𝑓) are close to each other – Much stricter notion than average sensitivity – Some of our algorithms inspired by differentially private algorithms ■ Stabi bilit lity y of f Learnin rning g Algo gorit rithms hms [Bousquet Elisseeff '02] – A learner is stable if empirical loss does not change much by replacing any sample in the training data – Stable learners have low generalization error 36

T alk Outline ■ Our definition of average sensitivity for graph algorithms ■ Key properties of our definition ■ Main results ■ Algorithm with low sensitivity for the global minimum cut problem ■ Conclusions and open directions 37

k-Average Sensitivity from Average Sensitivity Theorem: em: If 𝐵 has average sensitivity 𝑔(𝑜, 𝑛) , it has 𝑙 -average sensitivity at most σ 𝑗∈[𝑙] 𝑔(𝑜, 𝑛 − 𝑗 + 1) . 38

Average Sensitivity Composes Algorithms 𝐵, 𝐶, 𝐷 such that 𝐵(𝐻) = 𝐶(𝐻, 𝐷 𝐻 ) 39

Average Sensitivity Composes Algorithms 𝐵, 𝐶, 𝐷 such that 𝐵(𝐻) = 𝐶(𝐻, 𝐷 𝐻 ) Theorem em (Info formal) mal): : Average sensitivity of 𝐵 on 𝐻 = (𝑊, 𝐹) can be bounded by the sum of: a term for average sensitivity of 𝐶 , and • a term for average sensitivity of 𝐷 . • 40

Average Sensitivity Composes Algorithms 𝐵, 𝐶, 𝐷 such that 𝐵(𝐻) = 𝐶(𝐻, 𝐷 𝐻 ) Theorem em (Info formal) mal): : Average sensitivity of 𝐵 on 𝐻 = (𝑊, 𝐹) can be bounded by the sum of: a term for average sensitivity of 𝐶 , and • a term for average sensitivity of 𝐷 . • Can be used to bound the average sensitivity of a distribution over multiple stable-on-average algorithms. 41

Connection to Deterministic 𝐻 𝐵(𝐻) Sublinear Algorithms Algorithm 𝐵 42

Connection to Deterministic 𝐻 𝐵(𝐻) Sublinear Algorithms Algorithm 𝐵 Local simulator 𝑀 Graph 𝐻 43

Connection to Deterministic 𝐻 𝐵(𝐻) Sublinear Algorithms Algorithm 𝐵 1 if 𝑓 ∈ 𝐵(𝐻) Local 𝑓 ∈ 𝐹 simulator 𝑀 0, otherwise Graph 𝐻 44

Connection to Deterministic 𝐻 𝐵(𝐻) Sublinear Algorithms Algorithm 𝐵 𝑟 𝐻 ≜ 𝔽 𝑓∈𝐹 [# queries by 𝑀] 1 if 𝑓 ∈ 𝐵(𝐻) Local 𝑓 ∈ 𝐹 simulator 𝑀 0, otherwise Graph 𝐻 45

Connection to Deterministic 𝐻 𝐵(𝐻) Sublinear Algorithms Algorithm 𝐵 𝑟 𝐻 ≜ 𝔽 𝑓∈𝐹 [# queries by 𝑀] 1 if 𝑓 ∈ 𝐵(𝐻) Local 𝑓 ∈ 𝐹 simulator 𝑀 Avera erage ge se sensiti sitivit vity y of f 𝐵 0, otherwise on on 𝐻 is s ≤ 𝑟(𝐻) Graph 𝐻 46

𝜌 is the random string Connection to 𝐻 Algorithm 𝐵 𝐵 𝜌 (𝐻) Sublinear Algorithms 𝜌 𝑟 𝐻 ≜ 𝔽 𝜌,𝑓∈𝐹 [# queries by 𝑀] 𝑓 ∈ 𝐹 1 if 𝑓 ∈ 𝐵 𝜌 (𝐻) Local simulator 𝑀 Avera erage ge se sensiti sitivit vity y of f 𝐵 0, otherwise 𝜌 on on 𝐻 is s ≤ 𝑟(𝐻) Graph 𝐻 47

Graph problem 𝑄 Connection to Local Computation 1 if 𝑤 is part of LCA 𝑀 a solution to 𝑄 Algorithms (LCAs) 𝑤 ∈ 𝑊 on 𝐻 𝜌 ∈ 0,1 𝑠 0, otherwise Graph 𝐻 Answers of 𝑀 are consistent with a single feasible solution of 𝑄 on 𝐻 48

Graph problem 𝑄 Connection to Local Computation 1 if 𝑤 is part of LCA 𝑀 a solution to 𝑄 Algorithms (LCAs) 𝑤 ∈ 𝑊 on 𝐻 𝜌 ∈ 0,1 𝑠 0, otherwise If f a pr probl blem m 𝑄 has s an LCA of f qu query co comp mplexit xity y 𝑟(𝐻) , then it Graph 𝐻 has s an algo gorit ithm hm with avera rage ge sensi se sitivi ivity y ≤ 𝑟(𝐻) Answers of 𝑀 are consistent with a single feasible solution of 𝑄 on 𝐻 49

Graph problem 𝑄 Connection to Local Computation 1 if 𝑤 is part of LCA 𝑀 a solution to 𝑄 Algorithms (LCAs) 𝑤 ∈ 𝑊 on 𝐻 𝜌 ∈ 0,1 𝑠 0, otherwise If f a pr probl blem m 𝑄 has s an LCA of f qu query co comp mplexit xity y 𝑟(𝐻) , then it Graph 𝐻 has s an algo gorit ithm hm with avera rage ge se sensi sitivi ivity y ≤ 𝑟(𝐻) Answers of 𝑀 are consistent with a Lower r bo bound d on average ge single feasible solution of 𝑄 on 𝐻 sensi se sitivi ivity y imp mplies es lower er bo bound d on LCA A qu query y co comp mplexit xity! y! 50

T alk Outline ■ Our definition of average sensitivity for graph algorithms ■ Key properties of our definition ■ Main results ■ Algorithm with low sensitivity for the global minimum cut problem ■ Conclusions and open directions 51

Minimum Spanning Forest For graphs on 𝑜 vertices and 𝑛 edges Algo gorit rithm hm Avera erage ge Sensit nsitivi ivity Krusk skal' al's s Algo gorit rithm hm Prim' m's s Algo gorit rithm hm 52

Minimum Spanning Forest For graphs on 𝑜 vertices and 𝑛 edges Algo gorit rithm hm Avera erage ge Sensit nsitivi ivity Krusk skal' al's s Algo gorit rithm hm 𝑃(𝑜/𝑛) Prim' m's s Algo gorit rithm hm 53

Minimum Spanning Forest For graphs on 𝑜 vertices and 𝑛 edges Algo gorit rithm hm Avera erage ge Sensit nsitivi ivity Krusk skal' al's s Algo gorit rithm hm 𝑃(𝑜/𝑛) Prim' m's s Algo gorit rithm hm Ω(𝑜) For a specific tie- breaking rule 54

Other Problems We Study ■ Maximu mum m Cardi dina nality lity Match ching ng – Output an independent set of edges with maximum cardinality 55

Other Problems We Study ■ Maximu mum m Cardi dina nality lity Match ching ng – Output an independent set of edges with maximum cardinality ■ Globa bal l Minim imum um Cut – Output a subset 𝑇 of vertices with minimum number of edges between 𝑇 and 𝑊 ∖ 𝑇 56

Other Problems We Study ■ Maximu mum m Cardi dina nality lity Match ching ng – Output an independent set of edges with maximum cardinality ■ Globa bal l Minim imum um Cut – Output a subset 𝑇 of vertices with minimum number of edges between 𝑇 and 𝑊 ∖ 𝑇 ■ 𝑡 - 𝑢 Minimu mum m Cut ■ 2-Colo oloring ring 57

Maximum Cardinality Matching For graphs on 𝑜 vertices with max. matching size OP OPT 58

Maximum Cardinality Matching For graphs on 𝑜 vertices with max. matching size OP OPT App pproxima ximatio tion n Ra Ratio Avera erage ge Sensit nsitivi ivity 1 Ω(𝑜) 59

Maximum Cardinality Matching For graphs on 𝑜 vertices with max. matching size OP OPT App pproxima ximatio tion n Ra Ratio Avera erage ge Sensit nsitivi ivity 1 Ω(𝑜) 1/2 1 60

Maximum Cardinality Matching For graphs on 𝑜 vertices with max. matching size OP OPT App pproxima ximatio tion n Ra Ratio Avera erage ge Sensit nsitivi ivity 1 Ω(𝑜) 1/2 1 Corollar llary: 2-approximation algorithm for minimum vertex cover with average sensitivity 2 . 61

Maximum Cardinality Matching For graphs on 𝑜 vertices with max. matching size OP OPT App pproxima ximatio tion n Ra Ratio Avera erage ge Sensit nsitivi ivity 1 Ω(𝑜) 1/2 1 1 1+𝜗 2 𝑃𝑄𝑈 1 − 𝜗 𝑃 𝜗 3 Corollar llary: 2-approximation algorithm for minimum vertex cover with average sensitivity 2 . 62

Global Minimum Cut For graphs on 𝑜 vertices with global min. cut of size OP OPT 63

Global Minimum Cut For graphs on 𝑜 vertices with global min. cut of size OP OPT App pproxima ximatio tion n Ra Ratio Avera erage ge Sensiti nsitivit vity 1 Ω(𝑜) 64

Global Minimum Cut For graphs on 𝑜 vertices with global min. cut of size OP OPT App pproxima ximatio tion n Ra Ratio Avera erage ge Sensiti nsitivit vity 1 Ω(𝑜) 1 2 + 𝜗 𝑃( 𝜗 OPT ) 𝑜 65

Global Minimum Cut For graphs on 𝑜 vertices with global min. cut of size OP OPT App pproxima ximatio tion n Ra Ratio Avera erage ge Sensiti nsitivit vity 1 Ω(𝑜) 1 2 + 𝜗 𝑃( 𝜗 OPT ) 𝑜 If OP OPT = 𝜕(log 𝑜) , average sensitivity is 𝑃(1) 66

Global Minimum Cut For graphs on 𝑜 vertices with global min. cut of size OP OPT App pproxima ximatio tion n Ra Ratio Avera erage ge Sensiti nsitivit vity 1 Ω(𝑜) 1 2 + 𝜗 𝑃( 𝜗 OPT ) 𝑜 Ω(𝑜 1/ OPT / OPT 2 ) < ∞ If OP OPT = 𝜕(log 𝑜) , average sensitivity is 𝑃(1) 67

Global Minimum Cut For graphs on 𝑜 vertices with global min. cut of size OP OPT App pproxima ximatio tion n Ra Ratio Avera erage ge Sensiti nsitivit vity 1 Ω(𝑜) 1 2 + 𝜗 𝑃( 𝜗 OPT ) 𝑜 Ω(𝑜 1/ OPT / OPT 2 ) < ∞ If OP OPT = 𝜕(log 𝑜) , average sensitivity is 𝑃(1) If OP OPT = = O log 𝑜 , average sensitivity is (nearly) optimal 68

s-t Minimum Cut Probl blem em: Given graph 𝐻 and vertices 𝑡, 𝑢 , find output a subset 𝑇 of vertices with minimum number of edges between 𝑇 and 𝑊 ∖ 𝑇 such For graphs on 𝑜 vertices with s-t min. cut of size OP OPT that 𝑡 ∈ 𝑇 and 𝑢 ∈ 𝑊 ∖ 𝑇 App pproxima ximatio tion Avera erage ge Sensiti nsitivit vity (mu mult ltip iplic licative, ive, add dditive) e) (1, 𝑃(𝑜 2/3 )) 𝑃(𝑜 2/3 ) 69

2-Coloring Probl blem em: Given a bipartite graph 𝐻 , , output the set of vertices in one of the bipartitions. App pproxima ximatio tion Avera erage ge Sensiti nsitivit vity (mu mult ltip iplic licative, ive, add dditive) e) − Ω(𝑜) Every LCA for 2 -coloring has query complexity Ω(𝑜) Answers an open question raised by [Czumaj, Mansour, Vardi 18] on existence of sublinear-query LCAs for the problem of 2-coloring. 70

T alk Outline ■ Our definition of average sensitivity for graph algorithms ■ Key properties of our definition ■ Main results ■ Algorithm with low sensitivity for the global minimum cut problem ■ Conclusions and Open directions 71

Global Minimum Cut Problem Given 𝐻 = (𝑊, 𝐹) and 𝑇 ⊆ 𝑊 , size( 𝑇, 𝐻 ): number of edges crossing (𝑇, 𝑊 ∖ 𝑇) 72

Global Minimum Cut Problem Given 𝐻 = (𝑊, 𝐹) and 𝑇 ⊆ 𝑊 , size( 𝑇, 𝐻 ): number of edges crossing (𝑇, 𝑊 ∖ 𝑇) Probl blem: em: Output set 𝑇 ⊆ 𝑊 with the minimum size. 73

Global Minimum Cut Problem Given 𝐻 = (𝑊, 𝐹) and 𝑇 ⊆ 𝑊 , size( 𝑇, 𝐻 ): number of edges crossing (𝑇, 𝑊 ∖ 𝑇) Probl blem: em: Output set 𝑇 ⊆ 𝑊 with the minimum size. Polynomial time exact algorithms exist. 74

Global Minimum Cut Problem Given 𝐻 = (𝑊, 𝐹) and 𝑇 ⊆ 𝑊 , size( 𝑇, 𝐻 ): number of edges crossing (𝑇, 𝑊 ∖ 𝑇) Probl blem: em: Output set 𝑇 ⊆ 𝑊 with the minimum size. Polynomial time exact algorithms exist. Theorem em [Karge ger r 93]: : For 𝛽 ≥ 1 , the number of cuts of size at most 𝛽 ⋅ OPT is at most 𝑜 2𝛽 and they can be enumerated in polynomial time (per cut). 75

Global Minimum Cut Theorem em: : There exists a polynomial time (2 + 𝜗) -approximation algorithm with average sensitivity 1 𝑃 𝜗 OPT for the global minimum cut problem for all 𝜗 > 0 . 𝑜 76

Stable Algorithm for Global Minimum Cut On input 𝐻 = (𝑊, 𝐹) and parameter 𝜗 > 0 : • Compute the value OPT; log 𝑜 • Let 𝛽 ← 𝜄( 𝜗 OPT ) ; • Enumerate all cuts of size at most 2 + 𝜗 ⋅ OPT ; • Output a cut 𝑇 ⊆ 𝑊 with probability proportional to exp(−𝛽 ⋅ size 𝑇, 𝐻 ) 77

Stable Algorithm for Global Minimum Cut On input 𝐻 = (𝑊, 𝐹) and parameter 𝜗 > 0 : • Compute the value OPT; log 𝑜 • Let 𝛽 ← 𝜄( 𝜗 OPT ) ; • Enumerate all cuts of size at most 2 + 𝜗 ⋅ OPT ; • Output a cut 𝑇 ⊆ 𝑊 with probability proportional to exp(−𝛽 ⋅ size 𝑇, 𝐻 ) 78

Stable Algorithm for Global Minimum Cut On input 𝐻 = (𝑊, 𝐹) and parameter 𝜗 > 0 : • Compute the value OPT; log 𝑜 • Let 𝛽 ← 𝜄( 𝜗 OPT ) ; • Enumerate all cuts of size at most 2 + 𝜗 ⋅ OPT ; • Output a cut 𝑇 ⊆ 𝑊 with probability proportional to exp(−𝛽 ⋅ size 𝑇, 𝐻 ) 79

Stable Algorithm for Global Minimum Cut On input 𝐻 = (𝑊, 𝐹) and parameter 𝜗 > 0 : • Compute the value OPT; log 𝑜 • Let 𝛽 ← 𝜄( 𝜗 OPT ) ; • Enumerate all cuts of size at most 2 + 𝜗 ⋅ OPT ; • Output a cut 𝑇 ⊆ 𝑊 with probability proportional to exp(−𝛽 ⋅ size 𝑇, 𝐻 ) 80

Stable Algorithm for Global Minimum Cut Sa Samplin ling g from On input 𝐻 = (𝑊, 𝐹) and parameter 𝜗 > 0 : an approxim ximat ate e Gibb bbs s • Compute the value OPT; distribu tributio tion log 𝑜 • Let 𝛽 ← 𝜄( 𝜗 OPT ) ; • Enumerate all cuts of size at most 2 + 𝜗 ⋅ OPT ; • Output a cut 𝑇 ⊆ 𝑊 with probability proportional to exp(−𝛽 ⋅ size 𝑇, 𝐻 ) 81

Stable Algorithm for Global Minimum Cut Samplin Sa ling g from On input 𝐻 = (𝑊, 𝐹) and parameter 𝜗 > 0 : an approxim ximat ate e Gibb bbs s • Compute the value OPT; distribu tributio tion log 𝑜 • Let 𝛽 ← 𝜄( 𝜗 OPT ) ; • Enumerate all cuts of size at most 2 + 𝜗 ⋅ OPT ; • Output a cut 𝑇 ⊆ 𝑊 with probability proportional to exp(−𝛽 ⋅ size 𝑇, 𝐻 ) Inspired from a differentially private algorithm for global minimum cut [Gupta Ligett McSherry Roth T alwar '10] 82

Analysis App pproxima ximatio tion n Ra Ratio Clear from algorithm description Ru Runnin ning g time me Follows from Karger's theorem Avera erage ge Sensiti nsitivit vity Will analyze now 83

Analysis: A (Slightly) Different Algorithm On input 𝐻 = (𝑊, 𝐹) and parameter 𝜗 > 0 : Sa Samplin ling g from Gibb bbs s • Compute the value OPT; distrib tributio tion log 𝑜 • Let 𝛽 ← 𝜄( 𝜗 OPT ) ; • Output cut 𝑇 ⊆ 𝑊 with prob. proportional to exp(−𝛽 ⋅ size 𝑇, 𝐻 ) 84

Analysis: A (Slightly) Different Algorithm On input 𝐻 = (𝑊, 𝐹) and parameter 𝜗 > 0 : Sa Samplin ling g from Gibb bbs s • Compute the value OPT; distrib tributio tion log 𝑜 • Let 𝛽 ← 𝜄( 𝜗 OPT ) ; • Output cut 𝑇 ⊆ 𝑊 with prob. proportional to exp(−𝛽 ⋅ size 𝑇, 𝐻 ) Obse Ob servation ion: Enough to bound average sensitivity of above inefficient algorithm, since its output distribution is close to original algorithm 85

Analysis Denote the inefficient algorithm Overview using 𝐵 ■ Average sensitivity = Average On input 𝐻 = (𝑊, 𝐹) and (over 𝑓 ∈ 𝐹) earth mover's parameter 𝜗 > 0 : distance between 𝐵(𝐻) and 𝐵(𝐻 − 𝑓) • Compute the value OPT; log 𝑜 • Let 𝛽 ← 𝜄( 𝜗 OPT ) ; • Output cut 𝑇 ⊆ 𝑊 with prob. proportional to exp(−𝛽 ⋅ size 𝑇, 𝐻 ) 86

Analysis 𝑞 𝑇, 𝐻 :Probability that 𝐵 outputs cut 𝑇 Overview on input 𝐻 Fix 𝑓 ∈ 𝐹 . ■ For cuts 𝑇 such that 𝑓 crosses 𝑇 , 𝑞 𝑇, 𝐻 − 𝑓 ≈ 𝑞 𝑇, 𝐻 ⋅ exp 𝛽 On input 𝐻 = (𝑊, 𝐹) and parameter 𝜗 > 0 : ■ Earth mover's distance between 𝐵(𝐻) and 𝐵 𝐻 − 𝑓 • Compute the value OPT; log 𝑜 • Let 𝛽 ← 𝜄( 𝜗 OPT ) ; ≈ 𝑜 ⋅ ෍ 𝑞 𝑇, 𝐻 − 𝑓 − 𝑞 𝑇, 𝐻 • Output cut 𝑇 ⊆ 𝑊 with 𝑇:𝑓 crosses 𝑇 prob. proportional to = 𝑜 ⋅ exp 𝛽 − 1 ⋅ 𝑞 𝑇, 𝐻 ෍ exp(−𝛽 ⋅ size 𝑇, 𝐻 ) 𝑇:𝑓 crosses 𝑇 87

Analysis 𝑞 𝑇, 𝐻 :Probability that 𝐵 outputs cut 𝑇 Overview on input 𝐻 Fix 𝑓 ∈ 𝐹 . ■ For cuts 𝑇 such that 𝑓 crosses 𝑇 , 𝑞 𝑇, 𝐻 − 𝑓 ≈ 𝑞 𝑇, 𝐻 ⋅ exp 𝛽 On input 𝐻 = (𝑊, 𝐹) and parameter 𝜗 > 0 : ■ Earth mover's distance between 𝐵(𝐻) and 𝐵 𝐻 − 𝑓 • Compute the value OPT; log 𝑜 • Let 𝛽 ← 𝜄( 𝜗 OPT ) ; ≈ 𝑜 ⋅ ෍ 𝑞 𝑇, 𝐻 − 𝑓 − 𝑞 𝑇, 𝐻 • Output cut 𝑇 ⊆ 𝑊 with 𝑇:𝑓 crosses 𝑇 prob. proportional to = 𝑜 ⋅ exp 𝛽 − 1 ⋅ 𝑞 𝑇, 𝐻 ෍ exp(−𝛽 ⋅ size 𝑇, 𝐻 ) 𝑇:𝑓 crosses 𝑇 88

Analysis 𝑞 𝑇, 𝐻 :Probability that 𝐵 outputs cut 𝑇 Overview on input 𝐻 Fix 𝑓 ∈ 𝐹 . ■ For cuts 𝑇 such that 𝑓 crosses 𝑇 , 𝑞 𝑇, 𝐻 − 𝑓 ≈ 𝑞 𝑇, 𝐻 ⋅ exp 𝛽 On input 𝐻 = (𝑊, 𝐹) and parameter 𝜗 > 0 : ■ Earth mover's distance between 𝐵(𝐻) and 𝐵 𝐻 − 𝑓 • Compute the value OPT; log 𝑜 • Let 𝛽 ← 𝜄( 𝜗 OPT ) ; ≈ 𝑜 ⋅ ෍ 𝑞 𝑇, 𝐻 − 𝑓 − 𝑞 𝑇, 𝐻 • Output cut 𝑇 ⊆ 𝑊 with 𝑇:𝑓 crosses 𝑇 prob. proportional to = 𝑜 ⋅ exp 𝛽 − 1 ⋅ 𝑞 𝑇, 𝐻 ෍ exp(−𝛽 ⋅ size 𝑇, 𝐻 ) 𝑇:𝑓 crosses 𝑇 89

Analysis 𝑞 𝑇, 𝐻 :Probability that 𝐵 outputs cut 𝑇 Overview on input 𝐻 Fix 𝑓 ∈ 𝐹 . ■ For cuts 𝑇 such that 𝑓 crosses 𝑇 , 𝑞 𝑇, 𝐻 − 𝑓 ≈ 𝑞 𝑇, 𝐻 ⋅ exp 𝛽 On input 𝐻 = (𝑊, 𝐹) and parameter 𝜗 > 0 : ■ Earth mover's distance between 𝐵(𝐻) and 𝐵 𝐻 − 𝑓 • Compute the value OPT; log 𝑜 • Let 𝛽 ← 𝜄( 𝜗 OPT ) ; ≈ 𝑜 ⋅ ෍ 𝑞 𝑇, 𝐻 − 𝑓 − 𝑞 𝑇, 𝐻 • Output cut 𝑇 ⊆ 𝑊 with 𝑇:𝑓 crosses 𝑇 prob. proportional to = 𝑜 ⋅ exp 𝛽 − 1 ⋅ 𝑞 𝑇, 𝐻 ෍ exp(−𝛽 ⋅ size 𝑇, 𝐻 ) 𝑇:𝑓 crosses 𝑇 90

Analysis 𝑞 𝑇, 𝐻 :Probability that 𝐵 outputs cut 𝑇 Overview on input 𝐻 Fix 𝑓 ∈ 𝐹 . ■ For cuts 𝑇 such that 𝑓 crosses 𝑇 , 𝑞 𝑇, 𝐻 − 𝑓 ≈ 𝑞 𝑇, 𝐻 ⋅ exp 𝛽 On input 𝐻 = (𝑊, 𝐹) and parameter 𝜗 > 0 : ■ Earth mover's distance between 𝐵(𝐻) and 𝐵 𝐻 − 𝑓 • Compute the value OPT; log 𝑜 • Let 𝛽 ← 𝜄( 𝜗 OPT ) ; ≈ 𝑜 ⋅ ෍ 𝑞 𝑇, 𝐻 − 𝑓 − 𝑞 𝑇, 𝐻 • Output cut 𝑇 ⊆ 𝑊 with 𝑇:𝑓 crosses 𝑇 prob. proportional to = 𝑜 ⋅ exp 𝛽 − 1 ⋅ 𝑞 𝑇, 𝐻 ෍ exp(−𝛽 ⋅ size 𝑇, 𝐻 ) 𝑇:𝑓 crosses 𝑇 91

Analysis ■ Average sensitivity of 𝐵 is Overview ≈ 𝑜 𝑛 ⋅ exp 𝛽 − 1 ⋅ ෍ ෍ 𝑞 𝑇, 𝐻 𝑓 𝑇:𝑓 crosses 𝑇 ■ Average sensitivity of 𝐵 is On input 𝐻 = (𝑊, 𝐹) and 𝑜 parameter 𝜗 > 0 : 𝑛 ⋅ exp 𝛽 − 1 ⋅ (Expected ≤ size of cut output by 𝐵) • Compute the value OPT; ■ Expected size of cut log 𝑜 • Let 𝛽 ← 𝜄( 𝜗 OPT ) ; ≤ 2 + 𝜗 ⋅ OPT + 𝑝(1) • Output cut 𝑇 ⊆ 𝑊 with 2𝑛 ■ OPT ≤ 𝑜 , as min. cut size at most prob. proportional to average degree exp(−𝛽 ⋅ size 𝑇, 𝐻 ) (More Detailed Analysis Overview) 92

Analysis ■ Average sensitivity of 𝐵 is Overview ≈ 𝑜 𝑛 ⋅ exp 𝛽 − 1 ⋅ ෍ ෍ 𝑞 𝑇, 𝐻 𝑓 𝑇:𝑓 crosses 𝑇 ■ Average sensitivity of 𝐵 is On input 𝐻 = (𝑊, 𝐹) and 𝑜 parameter 𝜗 > 0 : 𝑛 ⋅ exp 𝛽 − 1 ⋅ (Expected ≤ size of cut output by 𝐵) • Compute the value OPT; ■ Expected size of cut log 𝑜 • Let 𝛽 ← 𝜄( 𝜗 OPT ) ; ≤ 2 + 𝜗 ⋅ OPT + 𝑝(1) • Output cut 𝑇 ⊆ 𝑊 with 2𝑛 ■ OPT ≤ 𝑜 , as min. cut size at most prob. proportional to average degree exp(−𝛽 ⋅ size 𝑇, 𝐻 ) (More Detailed Analysis Overview) 93

Analysis ■ Average sensitivity of 𝐵 is Overview ≈ 𝑜 𝑛 ⋅ exp 𝛽 − 1 ⋅ ෍ ෍ 𝑞 𝑇, 𝐻 𝑓 𝑇:𝑓 crosses 𝑇 ■ Average sensitivity of 𝐵 is On input 𝐻 = (𝑊, 𝐹) and 𝑜 parameter 𝜗 > 0 : 𝑛 ⋅ exp 𝛽 − 1 ⋅ (Expected ≤ size of cut output by 𝐵) • Compute the value OPT; ■ Expected size of cut log 𝑜 • Let 𝛽 ← 𝜄( 𝜗 OPT ) ; ≤ 2 + 𝜗 ⋅ OPT + 𝑝(1) • Output cut 𝑇 ⊆ 𝑊 with 2𝑛 ■ OPT ≤ 𝑜 , as min. cut size at most prob. proportional to average degree exp(−𝛽 ⋅ size 𝑇, 𝐻 ) (More Detailed Analysis Overview) 94

Analysis ■ Average sensitivity of 𝐵 is Overview ≈ 𝑜 𝑛 ⋅ exp 𝛽 − 1 ⋅ ෍ ෍ 𝑞 𝑇, 𝐻 𝑓 𝑇:𝑓 crosses 𝑇 ■ Average sensitivity of 𝐵 is On input 𝐻 = (𝑊, 𝐹) and 𝑜 parameter 𝜗 > 0 : 𝑛 ⋅ exp 𝛽 − 1 ⋅ (Expected ≤ size of cut output by 𝐵) • Compute the value OPT; ■ Expected size of cut log 𝑜 • Let 𝛽 ← 𝜄( 𝜗 OPT ) ; ≤ 2 + 𝜗 ⋅ OPT + 𝑝(1) • Output cut 𝑇 ⊆ 𝑊 with 2𝑛 ■ OPT ≤ 𝑜 , as min. cut size at most prob. proportional to average degree exp(−𝛽 ⋅ size 𝑇, 𝐻 ) (More Detailed Analysis Overview) 95

Global Minimum Cut Theorem em: : There exists a polynomial time (2 + 𝜗) -approximation algorithm with average sensitivity 1 𝑃 𝜗 OPT for the global minimum cut problem for all 𝜗 > 0 . 𝑜 96

Global Minimum Cut Theorem em: : There exists a polynomial time (2 + 𝜗) -approximation algorithm with average sensitivity 1 𝑃 𝜗 OPT for the global minimum cut problem for all 𝜗 > 0 . 𝑜 Samp Sa mplin ling g from m Gibb bbs s distri ributio ution gives es stabil ility ity 97

T alk Outline ■ Our definition of average sensitivity for graph algorithms ■ Key properties of our definition ■ Main results ■ Algorithm with low sensitivity for the global minimum cut problem ■ Conclusions and open directions 98

Summary of our contributions ■ Introduced a definition of sensitivity of graph algorithms with several useful properties 99

Summary of our contributions ■ Introduced a definition of sensitivity of graph algorithms with several useful properties ■ Design of stable algorithms for various combinatorial problems 100