Clique vs. Independent Set

Transcription

1 Lower Bounds for Clique vs. Independent Set Mika Göös University of Toronto Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 / 4

2 On page 6... Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 / 4

3 CIS problem [Yannakakis, STOC 88] G = ([n], E) Alice Bob Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 / 4

4 CIS problem [Yannakakis, STOC 88] G = ([n], E) Alice Clique x [n] of G Bob Independent set y [n] of G Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 / 4

5 CIS problem [Yannakakis, STOC 88] G = ([n], E) Alice Clique x [n] of G Bob Independent set y [n] of G Compute: CIS G (x, y) = x y Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 / 4

6 Background Yannakakis s motivation: Size of LPs for the vertex packing polytope of G Breakthrough: [Fiorini et al., STOC ] Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 / 4

7 Background Yannakakis s motivation: Size of LPs for the vertex packing polytope of G Breakthrough: [Fiorini et al., STOC ] Known bounds: G : NP cc (CIS G ) = log n (guess x y) Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 / 4

8 Background Yannakakis s motivation: Size of LPs for the vertex packing polytope of G Breakthrough: [Fiorini et al., STOC ] Known bounds: G : G : NP cc (CIS G ) = log n (guess x y) conp cc (CIS G ) O(log n) Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 / 4

9 Background Yannakakis s motivation: Size of LPs for the vertex packing polytope of G Breakthrough: [Fiorini et al., STOC ] Known bounds: G : G : NP cc (CIS G ) = log n (guess x y) conp cc (CIS G ) O(log n) Yannakakis s question: G : conp cc (CIS G ) O(log n)? Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 / 4

10 Background Alon Saks Seymour conjecture: G : χ(g) bp(g) +? Yannakakis s question: G : conp cc (CIS G ) O(log n)? Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 / 4

11 Background Alon Saks Seymour conjecture: G : χ(g) bp(g) +? [Huang Sudakov, ]: G : χ(g) bp(g) 6/5 Yannakakis s question: G : conp cc (CIS G ) O(log n)? Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 / 4

12 Background Polynomial Alon Saks Seymour conjecture: G : χ(g) poly( bp(g))? Yannakakis s question: G : conp cc (CIS G ) O(log n)? Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 / 4

13 Background Polynomial Alon Saks Seymour conjecture: G : χ(g) poly( bp(g))? [Alon Haviv] = = [Bousquet et al.] Yannakakis s question: G : conp cc (CIS G ) O(log n)? Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 / 4

14 Background Polynomial Alon Saks Seymour conjecture: G : χ(g) poly( bp(g))? [Alon Haviv] = = [Bousquet et al.] Yannakakis s question: G : conp cc (CIS G ) O(log n)? Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 / 4

15 Our result Main theorem G : conp cc (CIS G ) Ω(log.8 n) Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 4 / 4

16 Our result Main theorem G : conp cc (CIS G ) Ω(log.8 n) Prior bounds Measure Lower bound Reference P cc log n Kushilevitz, Linial, and Ostrovsky (999) conp cc 6/5 log n Huang and Sudakov () conp cc / log n Amano (4) conp cc log n Shigeta and Amano (4) Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 4 / 4

17 Our result Main theorem G : conp cc (CIS G ) Ω(log.8 n) Proof strategy: Query complexity Communication complexity Cf. lower bounds for log-rank [Nisan Wigderson, 995] Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 4 / 4

18 Models of communication F : X Y {, } Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 5 / 4

19 Models of communication NP cc Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 5 / 4

20 Models of communication UP cc Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 5 / 4

21 Models of communication CIS G is complete for UP cc : F CIS G UP cc (F) = UP cc (CIS G ) = log n Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 5 / 4

22 Proof strategy Restatement of Main theorem: F : X Y {, } conp cc (F) UP cc (F).8 Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 6 / 4

23 Proof strategy Restatement of Main theorem: F : X Y {, } conp cc (F) UP cc (F).8 Query separation: f : {, } n {, } conp dt ( f ) UP dt ( f ).8 Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 6 / 4

24 Proof strategy Restatement of Main theorem: F : X Y {, } conp cc (F) UP cc (F).8 Query separation: f : {, } n {, } conp dt ( f ) UP dt ( f ).8 Decision tree complexity measures: NP dt = DNF width = -certificate complexity conp dt = CNF width = -certificate complexity UP dt = Unambiguous DNF width Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 6 / 4

25 Proof strategy Restatement of Main theorem: F : X Y {, } conp cc (F) UP cc (F).8 Query separation: f : {, } n {, } conp dt ( f ) UP dt ( f ).8 Agenda: Step : Query separation Step : Simulation theorem [GLMWZ, 5] Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 6 / 4

26 Step : Query separation Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 7 / 4

27 Warm-up Example: Let f (x, x, x ) = iff x + x + x {, } UP dt ( f ) = because f x x x x x x conp dt ( f ) = because -input is fully sensitive Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 8 / 4

28 Warm-up Example: Let f (x, x, x ) = iff x + x + x {, } UP dt ( f ) = because f x x x x x x conp dt ( f ) = because -input is fully sensitive Recursive composition: f f ( ) := f ( ) f i+ ( ) := f ( f i ( ), f i ( ), f i ( )) x x x Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 8 / 4

29 Warm-up Example: Let f (x, x, x ) = iff x + x + x {, } UP dt ( f ) = because f x x x x x x conp dt ( f ) = because -input is fully sensitive Recursive composition: f f f f f ( ) := f ( ) f i+ ( ) := f ( f i ( ), f i ( ), f i ( )) x x x x 4 x 5 x 6 x 7 x 8 x 9 Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 8 / 4

30 Warm-up Example: Let f (x, x, x ) = iff x + x + x {, } UP dt ( f ) = because f x x x x x x conp dt ( f ) = because -input is fully sensitive Recursive composition: f f ( ) := f ( ) f i+ ( ) := f ( f i ( ), f i ( ), f i ( )) f x x x f x 4 x 5 x 6 f x 7 x 8 x 9 Hope: conp dt ( f i ) UP dt ( f i ) ( ) i Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 8 / 4

31 Warm-up Example: Problem! Let f (x, x, x ) = iff x + x + x {, } In order UP dt ( tof ) certify = because f i ( ) = f, x x x (should x x x be easy) might need conp dt to certify f ( f ) = because i ( ) = (should be hard) -input is fully sensitive Recursive composition: f f ( ) := f ( ) f i+ ( ) := f ( f i ( ), f i ( ), f i ( )) f x x x f x 4 x 5 x 6 f x 7 x 8 x 9 Hope: conp dt ( f i ) UP dt ( f i ) ( ) i Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 8 / 4

32 Warm-up Example: Problem! Let f (x, x, x ) = iff x + x + x {, } In order UP dt ( tof ) certify = because f i ( ) = f, x x x (should x x x be easy) might need conp dt to certify f ( f ) = because i ( ) = (should be hard) -input is fully sensitive Solution: Enlarge input/output alphabets Recursive composition: f x x x f f x 4 x 5 x 6 f : ({} Σ) n {} Σ f x 7 x 8 x 9 f ( ) := f ( ) f i+ ( ) := f ( f i ( ), f i ( ), f i ( )) Hope: conp dt ( f i ) UP dt ( f i ) ( ) i Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 8 / 4

33 Warm-up Example: Problem! Let f (x, x, x ) = iff x + x + x {, } In order UP dt ( tof ) certify = because f i ( ) = f, x x x (should x x x be easy) might need conp dt to certify f ( f ) = because i ( ) = (should be hard) -input is fully sensitive Solution: Enlarge input/output alphabets Recursive composition: f f : ({} Σ) n {} Σ f ( ) := f ( ) f i+ ( ) := f ( f i ( ), f i ( ), f i ( )) f f f Now: In order to certify f i ( ) = σ for σ Σ, only need to certify f i ( conp Hope: ) = σ for σ dt Σ( f i ) UP dt ( f x x x x 4 x 5 x 6 x 7 x 8 x i ) 9 ( ) i Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 8 / 4

34 Defining f Any two certificates in an UP dt decision tree intersect in variables = Finite projective planes! Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 9 / 4

35 Defining f Incidence ordering: Each node orders its incident edges using numbers from [] Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 9 / 4

36 Defining f Inputs to nodes: Pointer values from {} [] }{{} ( is a null pointer) =Σ Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 9 / 4

37 Defining f Defining f : ({} []) 7 {, } Say edge e is satisfied on input x iff all nodes v e point to e under x f (x) = iff x satisfies an edge Clearly UP dt ( f ) = Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 9 / 4

38 Defining f Problem! Certifying f ( ) = too easy! Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 9 / 4

39 Defining f Add input weights: f g 7 Gadget g is such that deciding if g( ) = i for i [] costs i queries Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 9 / 4

40 Defining f Add input weights: f g 7 Gadget g is such that deciding if g( ) = i for i [] costs i queries Else Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 9 / 4

41 Defining f Key properties: UP dt ( f g 7 ) = + + = 6 Certifying ( f g 7 )( ) = requires (# edges) = 7 queries Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 9 / 4

42 Defining f Key properties: UP dt ( f g 7 ) = + + = 6 Certifying ( f g 7 )( ) = requires (# edges) = 7 queries (Magic numerology: ) Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 9 / 4

43 Defining f Key properties: UP dt ( f g 7 ) = + + = 6 Recursive composition Certifying ( f g 7 )( ) = Key trick: requires (# edges) = 7 queries ) From ({} Σ) (Magic n {, } numerology: Construct ({} Σ) n {} {pointers} Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 9 / 4

44 Query separation: f : {, } n {, } conp dt ( f ) UP dt ( f ).8 Step : Simulation theorem from Rectangles Are Nonnegative Juntas Mika Göös, Shachar Lovett, Raghu Meka, Thomas Watson, and David Zuckerman (STOC 5) Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 / 4

45 Composed functions f g n f Compose with gn f z z z z 4 z 5 g g g g g x y x y x y x 4 y 4 x 5 y 5 Examples: Set-disjointness: OR AND n Inner-product: XOR AND n Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 / 4

46 Composed functions f g n f Compose with gn f z z z z 4 z 5 g g g g g x y x y x y x 4 y 4 x 5 y 5 Examples: In general: We choose: Set-disjointness: OR AND n Inner-product: XOR AND n g : {, } b {, } b {, } is a small gadget Alice holds x {, } bn Bob holds y {, } bn g = inner-product with b = Θ(log n) bits per party Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 / 4

47 Approximation by juntas Conical d-junta: Nonnegative combination of d-conjunctions (Example:.4 z z +.66 z z +.5 z z ) Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 / 4

48 Approximation by juntas Conical d-junta: Nonnegative combination of d-conjunctions (Example:.4 z z +.66 z z +.5 z z ) Main Structure Theorem: Suppose Π is cost-d randomised protocol for f g n Then there exists a conical d-junta h s.t. z dom f : Pr [ Π(x, y) accepts ] h(z) (x,y) (g n ) (z) Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 / 4

49 Approximation by juntas Conical d-junta: Nonnegative combination of d-conjunctions (Example:.4 z z +.66 z z +.5 z z ) Main Structure Theorem: Suppose Π is cost-d randomised protocol for f g n Then there exists a conical d-junta h s.t. z dom f : Pr [ Π(x, y) accepts ] h(z) (x,y) (g n ) (z) Cf. Polynomial approximation [Razborov, Sherstov, Shi Zhu,... ]: Approximate poly-degree of AND = Θ( n) Approximate junta-degree of AND = Θ(n) Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 / 4

50 Corollaries Simulation for NP: NP cc ( f g n ) = Θ(NP dt ( f ) b)... recall b = Θ(log n) Conical d-junta:.4 z z +.66 z z +.5 z z d-dnf: z z z z z z Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 / 4

51 Corollaries Simulation for NP: NP cc ( f g n ) = Θ(NP dt ( f ) b)... recall b = Θ(log n) Trivially: UP cc ( f g n ) O(UP dt ( f ) b) Main theorem follows! Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 / 4

52 Corollaries Simulation for NP: NP cc ( f g n ) = Θ(NP dt ( f ) b)... recall b = Θ(log n) NP MA Also covered! P BPP WAPP SBP PostBPP PP smooth rectangle corruption ext. discrepancy discrepancy Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 / 4

53 Summary Main result G : conp cc (CIS G ) Ω(log.8 n) Open problems Better separation for conp dt vs. UP dt? Simulation theorems for new models (e.g., BPP) Improve gadget size down to b = O() (Would give new proof of Ω(n) bound for set-disjointness) Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 4 / 4

54 Summary Main result G : conp cc (CIS G ) Ω(log.8 n) Open problems Better separation for conp dt vs. UP dt? Simulation theorems for new models (e.g., BPP) Improve gadget size down to b = O() (Would give new proof of Ω(n) bound for set-disjointness) Cheers! Mika Göös (Univ. of Toronto) Clique vs. Independent Set rd February 5 4 / 4