(Nearly) Efficient Algorithms for the Graph Matching Problem - PowerPoint PPT Presentation

(Nearly) Efficient Algorithms for the Graph Matching Problem Tselil Schramm (Harvard/MIT) with Boaz Barak, Chi-Ning Chou, Zhixian Lei & Yueqi Sheng (Harvard)

graph matching problem (approximate graph isomorphism) input: two graphs on 𝑜 vertices goal: find permutation of vertices that maximizes # shared edges ma𝑦 𝐵 𝐻 0 , 𝜌 𝐵 𝐻 1 𝜌 𝐻 1 𝐻 0

graph matching problem (approximate graph isomorphism) input: two graphs on 𝑜 vertices goal: find permutation of vertices that maximizes # shared edges ma𝑦 𝐵 𝐻 0 , 𝜌 𝐵 𝐻 1 𝜌 # matched = 4 𝐻 1 𝐻 0

graph matching problem (approximate graph isomorphism) input: two graphs on 𝑜 vertices goal: find permutation of vertices that maximizes # shared edges ma𝑦 𝐵 𝐻 0 , 𝜌 𝐵 𝐻 1 𝜌 𝐻 1 𝐻 0

graph matching problem (approximate graph isomorphism) input: two graphs on 𝑜 vertices goal: find permutation of vertices that maximizes # shared edges ma𝑦 𝐵 𝐻 0 , 𝜌 𝐵 𝐻 1 𝜌 # matched = 5 𝐻 1 𝐻 0

computationally hard (of course) NP-hard: reduction from quadratic assignment problem (non-simple graphs). [ Lawler’63] also: reduction from sparse random 3-SAT to approximate version [ O’Donnell -Wright-Wu- Zhou’14]

practitioners: undeterred  computational biology [e.g. Singh-Xu- Berger‘08]  de-anonymization [e.g. Narayanan- Shmatikov’09]  social networks [e.g. Korula- Lattanzi’14]  image alignment [e.g. Cho- Lee’12]  machine learning [e.g. Cour-Srinivasan- Shi’07]  pattern recognition, e.g. “thirty years of graph matching in pattern recognition” [Conte-Foggia-Sansone- Vento’04 ]

“robust average - case graph isomorphism” average case: correlated random graphs structured model sample 𝐻 ∼ 𝐻(𝑜, 𝑞) 𝐻 ≈ 𝑞𝛿 2 ⋅ 𝑜 𝛿 𝛿 ma𝑦 𝐵 𝐻 0 , 𝜌 𝐵 𝐻 1 2 𝜌 subsample edges w/prob 𝛿 𝐻 0 𝐻 1 avg. degree 𝑞𝛿 ⋅ 𝑜 random permutation 𝜌 [e.g. Pedarsani- Grossglauser’11, 𝐻 0 Lyzinski-Fishkind- Priebe’14, Korula- Lattanzi’14]

“robust average - case graph isomorphism” average case: correlated random graphs structured model “null” model sample 𝐻 ∼ 𝐻(𝑜, 𝑞) 𝐻 𝛿 𝛿 subsample edges w/prob 𝛿 𝐻 0 𝐻 1 avg. degree 𝑞𝛿 ⋅ 𝑜 𝜌 ≈ 𝑞𝛿 2 ⋅ 𝑜 𝐻 0 2

“robust average - case graph isomorphism” average case: correlated random graphs structured model “null” model sample sample 𝐻 0 , 𝐻 1 ∼ 𝐻(𝑜, 𝑞𝛿) 𝐻 ∼ 𝐻(𝑜, 𝑞) 𝐻 𝛿 𝛿 subsample edges w/prob 𝛿 𝐻 1 𝐻 0 𝐻 0 𝐻 1 avg. degree 𝑞𝛿 ⋅ 𝑜 avg. degree 𝑞𝛿 ⋅ 𝑜 𝜌 ≈ 𝑞𝛿 2 ⋅ 𝑜 ≈ 𝑞𝛿 2 ⋅ 𝑜 ma𝑦 𝐵 𝐻 0 , 𝜌 𝐵 𝐻 1 𝐻 0 2 𝜌 2

information theoretic limit 𝐻(𝑜, 𝑞) 𝐻 𝛿 𝛿 for which 𝑞, 𝛿 can we recover 𝜌 ? 𝐻 0 𝐻 1 𝜌 𝐻 0 Theorem [Cullina- Kivayash’16&17] Iff 𝑞𝛿 2 > log 𝑜 𝑜 , with high probability 𝜌 is the unique maximizing permutation.

algorithms for robust average case? average-case graph isomorphism algorithms fail. e.g. matching local neighborhoods match radius- 𝑙 neighborhoods? 𝐻 1 𝐻 0

algorithms for robust average case? average-case graph isomorphism algorithms fail. e.g. matching local neighborhoods 𝛿 𝛿 match radius- 𝑙 neighborhoods? 𝐻 1 𝐻 0

algorithms for robust average case? average-case graph isomorphism algorithms fail. e.g. spectral algorithm 𝑤 max 𝑤 max unique entries in top eigenvector give isomorphism? 𝐻 1 𝐻 0

algorithms for robust average case? average-case graph isomorphism algorithms fail. e.g. spectral algorithm 𝛿 𝛿 𝛿 𝛿 𝑤 max 𝑤 max + + unique entries in top eigenvector give isomorphism? 𝐻 1 𝐻 0 perturb eigenvectors by ≈ 𝛿

actual algorithms for robust average case?

starting from a seed 𝜌ȁ 𝑇 known 𝑇 𝜌(𝑇) 𝐻 0 𝐻 1

starting from a seed match vertices with similar adjacency into 𝑇 𝑤 𝑣 𝐻 1 𝐻 0 𝑇

starting from a seed match vertices with similar adjacency into 𝑇 𝜌 𝑣 = 𝑤 𝐻 1 𝐻 0 𝑇 iff seed ≥ Ω(𝑜 𝜗 ) , the seeded algorithm approximately recovers 𝜌 . [Yartseva- Grossglauser’13] 𝑃 𝑜 𝜗 time to guess a seed. need 2 ෨

𝐻(𝑜, 𝑞) structured 𝐻 our results 𝛿 𝛿 𝐻 0 𝐻 1 𝜌 Theorem 𝐻 0 1 2 𝑜 𝑝(1) 𝑜 1−𝜗 𝑜 153 𝑜 3 and 𝛿 = Ω(1) ,* there is a 𝑜 𝑃(log 𝑜) For any 𝜗 > 0 , if 𝑞 ∈ , ∪ 𝑜 , 𝑜 𝑜 𝑜 time algorithm that recovers 𝜌 on 𝑜 − 𝑝(𝑜) of the vertices w/prob ≥ 0.99 . 1 *we allow 𝛿 = Ω loglog 𝑜 𝑜 1/2 𝑜 𝑝(1) 𝑜 1/153 𝑜 2/3 𝑜 1−𝜗 𝐻(𝑜, 𝑞) log 𝑜 𝑜 average degrees: 𝑜 3/5 𝑜 1/3

𝐻(𝑜, 𝑞) structured 𝐻 our results 𝛿 𝛿 𝐻 0 𝐻 1 𝜌 Theorem 𝐻 0 1 2 𝑜 𝑝(1) 𝑜 1−𝜗 𝑜 153 𝑜 3 and 𝛿 = Ω(1) ,* there is a 𝑜 𝑃(log 𝑜) For any 𝜗 > 0 , if 𝑞 ∈ , ∪ 𝑜 , 𝑜 𝑜 𝑜 time algorithm that recovers 𝜌 on 𝑜 − 𝑝(𝑜) of the vertices w/prob ≥ 0.99 . 1 *we allow 𝛿 ≥ log 𝑝(1) 𝑜 Theorem 𝐻(𝑜, 𝑞𝛿) hypothesis testing If 𝑞, 𝛿 are as above then there is a poly(𝑜) time distinguishing null algorithm for the structured vs null distributions. 𝐻 0 𝐻 1

our approach: small subgraphs hypothesis testing: correlation of subgraph counts recovery: match rare subgraphs seedless algorithms!

outline  distinguishing/hypothesis testing  recovery  concluding

outline  distinguishing/hypothesis testing  recovery  concluding

distinguishing/hypothesis testing structured 𝐻(𝑜, 𝑞) 𝐻 Given 𝐻 0 , 𝐻 1 sampled equally likely from structured or null , 𝛿 𝛿 decide w/prob 1 − 𝑝(1) from which. 𝐻 0 𝐻 1 ? ? 𝜌 𝐻 0 𝐻 1 𝐻 0 ? 𝐻(𝑜, 𝑞𝛿) null brute force: is there a 𝜌 with ≥ 𝑞𝛿 2 𝑜 2 matched edges? 𝐻 0 𝐻 1

…counting triangles? structured 𝐻(𝑜, 𝑞) 𝑑𝑝𝑠 𝐿 3 𝐻 0 , 𝐻 1 : = # 𝐿 3 𝑗𝑜 𝐻 0 # K 3 𝑗𝑜 𝐻 1 . 𝐻 𝛿 𝛿 𝐻 0 𝐻 1 ? ? 𝜌 𝐻 0 𝐻 1 𝐻 0 ? 𝐻(𝑜, 𝑞𝛿) null 𝐻 0 𝐻 1

…counting triangles? structured 𝐻(𝑜, 𝑞) 𝑑𝑝𝑠 𝐿 3 𝐻 0 , 𝐻 1 : = # 𝐿 3 𝑗𝑜 𝐻 0 # K 3 𝑗𝑜 𝐻 1 . 𝐻 𝛿 𝛿 𝐻 0 𝐻 1 𝜌 𝐻 0 𝐻 1 𝐻 0 𝐻(𝑜, 𝑞𝛿) null triangle counts in 𝐻 0 , 𝐻 1 are independent 𝐻 0 𝐻 1 𝑙 3 (𝐻 0 , 𝐻 1 ) ≈ 𝑞𝛿𝑜 6 𝔽 𝑑𝑝𝑠

…counting triangles? structured 𝐻(𝑜, 𝑞) 𝐿 3 (𝐻 0 , 𝐻 1 ) ≈ 𝑞𝛿𝑜 6 + 𝛿 2 𝑞𝑜 3 𝔽 𝑑𝑝𝑠 𝐻 𝛿 𝛿 triangle counts in 𝐻 0 , 𝐻 1 are correlated 𝐻 0 𝐻 1 𝜌 𝐻 0 𝐻 1 𝐻 0 𝐻(𝑜, 𝑞𝛿) null 𝐻 0 𝐻 1

…counting triangles? structured 𝐻(𝑜, 𝑞) 𝐻 𝛿 𝛿 structured 𝐿 3 (𝐻 0 , 𝐻 1 ) ≈ 𝑞𝛿𝑜 6 + 𝛿 2 𝑞𝑜 3 𝔽 𝑑𝑝𝑠 𝐻 0 𝐻 1 𝜌 null 𝐿 3 (𝐻 0 , 𝐻 1 ) ≈ 𝑞𝛿𝑜 6 𝔽 𝑑𝑝𝑠 𝐻 0 Variance? 𝐻(𝑜, 𝑞𝛿) null Optimistically, in null case, 1/2 ≈ 𝑞𝛿𝑜 3 𝕎 𝑑𝑝𝑠 𝐿 3 𝐻 0 , 𝐻 1 𝐻 0 𝐻 1

“independent trials” 𝑈 𝑑𝑝𝑠 𝑈 𝐻 0 , 𝐻 1 = 1 (𝑗) (𝐻 0 , 𝐻 1 ) 𝑈 ෍ 𝑑𝑝𝑠 Suppose we had 𝑈 “independent trials”: 𝐿 3 𝑗=1 𝐻(𝑜, 𝑞) 𝐻 𝛿 structured 𝛿 𝐻 0 𝐻 1 ≈ 𝑞𝛿𝑜 6 + 𝛿 2 𝑞𝑜 3 𝔽 𝑑𝑝𝑠 𝑈 𝐻 0 , 𝐻 1 𝜌 𝐻 0 if 𝑈 > 1/𝛿 6 , ≈ 𝑞𝛿𝑜 6 𝔽 𝑑𝑝𝑠 𝑈 𝐻 0 , 𝐻 1 𝐻(𝑜, 𝑞𝛿) 𝑑𝑝𝑠 𝑈 is a good test null 1/2 ≈ 1 𝑞𝛿𝑜 3 𝕎 𝑑𝑝𝑠 𝑈 𝐻 0 , 𝐻 1 𝐻 0 𝐻 1 𝑈

near-independent subgraphs “independent trials” 𝑈 𝑑𝑝𝑠 𝑈 𝐻 0 , 𝐻 1 = 1 Suppose we had 𝑈 “independent” subgraphs : 𝑈 ෍ 𝑑𝑝𝑠 𝐼 𝑗 (𝐻 0 , 𝐻 1 ) 𝑗=1 𝐼 1 , … , 𝐼 𝑈 what properties must 𝐼 1 , … , 𝐼 𝑈 have to be “independent”?

surprisingly delicate (concentration) 𝑞 = 𝑜 −5/7 𝐻(𝑜, 𝑞) 𝐻 𝐼 How many labeled copies of 𝐼 in 𝐻 ? ȁ𝑏𝑣𝑢 𝐼 ȁ ⋅ 𝑜 5! 5 ⋅ 𝑞 7 ≈ 𝑜 5 𝑞 7 = Θ(1) 𝔽 # 𝐼 𝐻 =

surprisingly delicate (concentration) 𝑞 = 𝑜 −5/7 𝐻(𝑜, 𝑞) 𝐻 𝐼 # 𝐼 (𝐻) does not concentrate! How many labeled copies of 𝐼 in 𝐻 ? ȁ𝑏𝑣𝑢 𝐼 ȁ ⋅ 𝑜 5! 5 ⋅ 𝑞 7 ≈ 𝑜 5 𝑞 7 = Θ(1) 𝔽 # 𝐼 𝐻 = How many labeled copies of 𝐿 4 in 𝐻 ? ȁ𝑏𝑣𝑢 𝐿 4 ȁ ⋅ 𝑜 4! 4 ⋅ 𝑞 6 ≈ 𝑜 4 𝑞 6 = Θ(𝑜 −2/7 ) 𝔽 # 𝐿 4 𝐻 =

variance of subgraph counts 𝐻(𝑜, 𝑞) 𝐻 𝐼 Lemma For a constant-sized subgraph 𝐼 , 2 𝔽 # 𝐼 𝐻 𝕎 # 𝐼 (𝐻) = Θ 1 ⋅ min 𝐾⊂𝐼 𝔽[# 𝐾 𝐻 ] subgraph of 𝐼 with fewest expected appearances