Homomorphism Testing

Transcription

1 Homomorphism Shpilka & Wigderson, 2006 Daniel Shahaf Tel-Aviv University January 2009

2 Table of Contents / 26

3 Algebra Definitions 1 Algebra Definitions 2 / 26

4 Groups Algebra Definitions Definition A group is a set G and a binary function + : G 2 G such that + has identity: g + 0 = 0 + g = g. inverses: g + g = g + g = 0. associativity: (g + g ) + g = g + (g + g ). If + is commutative, then (G, +) is called Abelian. 3 / 26

5 Groups Algebra Definitions Definition A group is a set G and a binary function + : G 2 G such that + has identity: g + 0 = 0 + g = g. inverses: g + g = g + g = 0. associativity: (g + g ) + g = g + (g + g ). If + is commutative, then (G, +) is called Abelian. Notational convention In this work, G is a group written additively and H is a group written multiplicatively. 3 / 26

6 (Affine) Homomorphisms Algebra Definitions Definition A homomorphism from a group G to a group H is a function f : G H such that f (g 1 ) f (g 2 ) = f (g 1 + g 2 ) for any g 1, g 2 G. Definition If f = h f and f Hom(G, H ), then f is an affine homomorphism. Is f h necessarily an affine homomorphism? 4 / 26

7 (Affine) Homomorphisms Algebra Definitions Definition A homomorphism from a group G to a group H is a function f : G H such that f (g 1 ) f (g 2 ) = f (g 1 + g 2 ) for any g 1, g 2 G. Definition If f = h f and f Hom(G, H ), then f is an affine homomorphism. Is f h necessarily an affine homomorphism? Lemma (Exercise) A function f is an affine homomorphism the function f := x f (0) 1 f (x) is a homomorphism. 4 / 26

8 Distances Algebra Definitions Definition The (normalized) Hamming distance of two functions f, g : D R is the fraction of D on which they disagree: dist(f, g) := Prob [f (x) g(x)]. x D Definition We say that a function f : G H is ɛ-close to linearity if dist(f, Hom(G, H )) ɛ. Otherwise f is said to be ɛ-far. 5 / 26

9 Motivation Formalization Past Answers 2 Motivation Formalization Past Answers 6 / 26

10 Linear Functions Motivation Formalization Past Answers Many problems can be presented or solved in terms of questions about functions and linearity: Numerical computations. The PCP theorem. Dictator functions. 7 / 26

11 Linear Functions Motivation Formalization Past Answers Many problems can be presented or solved in terms of questions about functions and linearity: Numerical computations. The PCP theorem. Dictator functions. Questions Is this function linear? How little need I change this function to make it linear? How do I change this function to make it linear? 7 / 26

12 Defining Motivation Formalization Past Answers Question Given a function f : G H, is f a homomorphism, or ɛ-far from linearity? 8 / 26

13 Defining Motivation Formalization Past Answers Question Given a function f : G H, is f a homomorphism, or ɛ-far from linearity? Yet another gap problem. Oracle-access model. 8 / 26

14 Defining The Solutions Motivation Formalization Past Answers We will seek probabilistic tests that classify functions with high accuracy. The tests will accept all valid inputs with probability 1. Need to show that they reject invalid (ɛ-far) inputs with high probability. 9 / 26

15 Defining The Solutions Motivation Formalization Past Answers We will seek probabilistic tests that classify functions with high accuracy. The tests will accept all valid inputs with probability 1. Need to show that they reject invalid (ɛ-far) inputs with high probability. A (δ, ɛ)-test for a property P D is a algorithm(d) that, on input that is ɛ-far from having P, accepts with probability at most δ. 9 / 26

16 Defining The Solutions Motivation Formalization Past Answers We will seek probabilistic tests that classify functions with high accuracy. The tests will accept all valid inputs with probability 1. Need to show that they reject invalid (ɛ-far) inputs with high probability. A (δ, ɛ)-test for a property P D is a algorithm(d) that, on input that is ɛ-far from having P, accepts with probability at most δ. Properties of tests Query complexity. Probability distance relation. Asymptotical running time. Number of random bits. 9 / 26

17 The BLR Test Motivation Formalization Past Answers test Pick at random x, y G. Accept iff f (x) f (y) = f (x + y). 10 / 26

18 The BLR Test Motivation Formalization Past Answers test Pick at random x, y G. Accept iff f (x) f (y) = f (x + y). If f Hom(G, H ), the test passes with probability 1. If the test passes with high probability, then f is ɛ-close. The BLR test is a (δ, δ/3 + O(δ 2 )) test. }{{} ɛ 10 / 26

19 The BLR Test Motivation Formalization Past Answers test Pick at random x, y G. Accept iff f (x) f (y) = f (x + y). If f Hom(G, H ), the test passes with probability 1. If the test passes with high probability, then f is ɛ-close. The BLR test is a (δ, δ/3 + O(δ 2 )) test. }{{} ɛ Exercise Extend the BLR linearity test to affine homomorphisms. 10 / 26

20 Graph Tests Motivation Formalization Past Answers The BLR test picked random x, y G. Generalization (Graph tests) Fix E G 2. E is viewed as a graph over vertices V = G. BLR: E = G 2 Pick random (x, y) E. 11 / 26

21 Graph Tests Motivation Formalization Past Answers The BLR test picked random x, y G. Generalization (Graph tests) Fix E G 2. E is viewed as a graph over vertices V = G. BLR: E = G 2 Pick random (x, y) E. Question Which graphs induce good tests? 11 / 26

22 Good Graphs Motivation Formalization Past Answers Problem: the BLR test requires 2 log G random bits. New goal: minimize randomness. Lower bound: log G + O(1). 12 / 26

23 Good Graphs Motivation Formalization Past Answers Problem: the BLR test requires 2 log G random bits. New goal: minimize randomness. Lower bound: log G + O(1). Idea: use graph tests. The graph test T E (induced by E G) requires only log E random bits. 12 / 26

24 Good Graphs Motivation Formalization Past Answers Problem: the BLR test requires 2 log G random bits. New goal: minimize randomness. Lower bound: log G + O(1). Idea: use graph tests. The graph test T E (induced by E G) requires only log E random bits. Theorem For all but e G -fraction of graphs E sized C G log H, and for every δ > 0, the test T E is a (δ, δ/3 + O(δ 2 ) + e G )-test for linearity. Need log G + log log H + O(1) randomness. Problem: which explicit such graphs are good/bad? 12 / 26

25 Proposed Test 3 Proposed Test 13 / 26

26 Cayley Graphs Proposed Test Let S H be a generating subset. Definition The Cayley graph G = Cay(H ; S) is the graph over vertices H with edges { h, h s h H, s S }. 14 / 26

27 Cayley Graphs Proposed Test Let S H be a generating subset. Definition The Cayley graph G = Cay(H ; S) is the graph over vertices H with edges { h, h s h H, s S }. Properties S -regular. If S is symmetric then G is undirected. G is vertex-transitive: u, v G h u, h v G. G is simple iff 1 / S. 14 / 26

28 Eigenvalues Proposed Test Definition The eigenvalues of a graph G are the eigenvalues of its adjacency matrix (M ij = [ i, j G]). Facts about Cayley graphs The top eigenvalue is λ 1 = d. 15 / 26

29 Eigenvalues Proposed Test Definition The eigenvalues of a graph G are the eigenvalues of its adjacency matrix (M ij = [ i, j G]). Facts about Cayley graphs The top eigenvalue is λ 1 = d. Define the normalized second eigenvalue: then λ 1 (i.e., λ 2 λ 1 ). λ[g] := 1 d max(λ 2, λ n ). If we remove any 2δdn edges from the graph, then remains a connected component of size at least (1 ɛ)n. ɛ := 4δ/ (1 λ[g]) < 1/3. 15 / 26

30 Constructive Tests Proposed Test Definition For G = Cay(G; S), let T G be the test that picks at random an edge x, x + s Cay(G; S) and checks whether f (x) f (s) = f (x + s). 16 / 26

31 Constructive Tests Proposed Test Definition For G = Cay(G; S), let T G be the test that picks at random an edge x, x + s Cay(G; S) and checks whether f (x) f (s) = f (x + s). Theorem For every G, H, and symmetric subset S G, the test T G accepts with probability 1 all homomorphisms h : G H ; where rejects with probability δ any function f : G H which is ɛ-far from being an affine homomorphism; δ := Prob [f (g) f (s) f (g + s)]; ɛ := 4δ/ (1 λ[g]) < 1/3. 16 / 26

32 Extracting a Homomorphism Take f, G, H, S, G, T G, δ, ɛ < 1/3, and λ := λ[g] as before. Denote d := S and n := G. Proposed Test Conclusion If δ ɛ is small, then f is close to an affine homomorphism. 17 / 26

33 Extracting a Homomorphism Proposed Test Take f, G, H, S, G, T G, δ, ɛ < 1/3, and λ := λ[g] as before. Denote d := S and n := G. Overview 1 For every x, almost all y agree on the value of [ ϕ(x) := Plurality f (x + y) f (y) 1 ]. y G 2 ϕ is a homomorphism. 3 ϕ is close to an affine shift of f. Conclusion If δ ɛ is small, then f is close to an affine homomorphism. 17 / 26

34 The Plurality Function I Proposed Test Claim For every x G, [ Prob ϕ(x) = f (x + y) f (y) 1] 1 ɛ. y G Proof go back skip proof 18 / 26

35 The Plurality Function I Proposed Test Claim For every x G, [ Prob ϕ(x) = f (x + y) f (y) 1] 1 ɛ. y G Proof Fix x G. By definition, δ = Prob [f (x + y)f (s) f (x + y + s)]. y G,s S Number of bad edges, y, y + s G having f (y) f (s) f (y + s) or f (x + y)f (s) f (x + y + s), is at most 2δdn. 18 / 26

36 The Plurality Function II Proposed Test Let H x G be obtained by removing all (undirected) bad edges, and let C x H x be a connected component of size (1 ɛ)n. Exists because there are at most 2δdn bad edges. 19 / 26

37 The Plurality Function II Proposed Test Let H x G be obtained by removing all (undirected) bad edges, and let C x H x be a connected component of size (1 ɛ)n. Exists because there are at most 2δdn bad edges. Lemma For any u, v C x, we have f (x + v) f (v) 1 = f (x + u) f (u) 1. Proof Take a path v = v 1,..., v t = u in C x. Write v i+1 = v i, v i + s i. Observe that f (v i ) 1 f (v i+1 ) = f (s i ) = f (x + v i ) 1 f (x + v i+1 ). 19 / 26

38 The Plurality Function III Proposed Test Conclusion Thus, f (x + v) f (v) 1 is constant for v C x. C x (1 ɛ)n > 1 2 G. 20 / 26

39 The Homomorphism I Claim ϕ is a homomorphism. Proposed Test Proof go back skip proof 21 / 26

40 The Homomorphism I Proposed Test Claim ϕ is a homomorphism. Proof Fix x, y G. Define: Then: p := Prob [ϕ(x) ϕ(y) = ϕ(x + y)]. h G A(x, y) := { ϕ(x) = f (y) f ( x + y) 1 }. p { 0, 1 }. p Prob [A(x, x + h) A(y, h) A(x + y, x + h)]. h G 21 / 26

41 The Homomorphism II Proposed Test From Claim 1: Prob [A(x, x + h)] 1 ɛ; h G [ ( )] Prob [A(y, h)] = Prob A y, y + ( y + h) h G h G [ = Prob A(y, y + h ) ] h G 1 ɛ; Prob [A(x + y, x + h)] 1 ɛ. h G 22 / 26

42 The Homomorphism II From Claim 1: Proposed Test Prob [A(x, x + h)] h G Prob [A(y, h)] h G Prob [A(x + y, x + h)] h G 1 ɛ 1 ɛ 1 ɛ Therefore: p Prob h G [A(x, x + h) A(y, h) A(x + y, x + h)]. p 1 3ɛ > / 26

43 The Homomorphism II From Claim 1: Proposed Test Prob [A(x, x + h)] h G Prob [A(y, h)] h G Prob [A(x + y, x + h)] h G 1 ɛ 1 ɛ 1 ɛ Therefore: p Prob [A(x, x + h) A(y, h) A(x + y, x + h)]. h G p 1 3ɛ > 0. p { 0, 1 }. p = 1. ϕ is a homomorphism. 22 / 26

44 The Affine Shift Claim There is some γ H such that dist(ϕ, f γ) ɛ. Proposed Test Proof go back skip proof 23 / 26

45 The Affine Shift Proposed Test Claim There is some γ H such that dist(ϕ, f γ) ɛ. Proof G x,y := { ϕ(x) = f (x + y) f (y) 1 } x. { y : G x,y } C x (1 ɛ) G 23 / 26

46 The Affine Shift Proposed Test Claim There is some γ H such that dist(ϕ, f γ) ɛ. Proof G x,y := { ϕ(x) = f (x + y) f (y) 1 } x. { y : G x,y } C x (1 ɛ) G x. Prob [G x,y ] 1 ɛ y Prob [G x,y] 1 ɛ x,y y 0. Prob [G x,y0 ] 1 ɛ x 23 / 26

47 The Affine Shift Proposed Test Claim There is some γ H such that dist(ϕ, f γ) ɛ. Proof G x,y := { ϕ(x) = f (x + y) f (y) 1 } x. { y : G x,y } C x (1 ɛ) G x. Prob [G x,y ] 1 ɛ y Prob [G x,y] 1 ɛ x,y y 0. Prob [G x,y0 ] 1 ɛ x [ ϕ(x) = f (x + y0 ) f (y 0 ) 1] 1 ɛ Prob x [ Prob ϕ(x + ( y 0 )) = f (x ) f (y 0 ) 1] 1 ɛ x G 23 / 26

48 Extracting a Homomorphism Proposed Test Take f, G, H, S, G, T G, δ, ɛ < 1/3, and λ := λ[g] as before. Denote d := S and n := G. Overview 1 For every x, almost all y agree on the value of [ ϕ(x) := Plurality f (x + y) f (y) 1 ]. y G 2 ϕ is a homomorphism. 3 ϕ is close to an affine shift of f. Conclusion If δ ɛ is small, then f is close to an affine homomorphism. 24 / 26

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50 The answer is soup. The End.