Gadgets and Anti-Gadgets Leading to a Complexity Dichotomy Tyson - - PowerPoint PPT Presentation

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Gadgets and Anti-Gadgets Leading to a Complexity Dichotomy Tyson - - PowerPoint PPT Presentation

May 06, 2023 371 likes •1.06k views

Gadgets and Anti-Gadgets Leading to a Complexity Dichotomy Tyson Williams University of Wisconsin-Madison Joint with: Jin-Yi Cai (University of Wisconsin-Madison) Michael Kowalczyk (Northern Michigan University) To appear at ITCS 2012 Tyson

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Gadgets and Anti-Gadgets Leading to a Complexity Dichotomy

Tyson Williams University of Wisconsin-Madison Joint with: Jin-Yi Cai (University of Wisconsin-Madison) Michael Kowalczyk (Northern Michigan University) To appear at ITCS 2012

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 1 / 19

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#VertexCover

Definition

A vertex cover of a graph is a set of vertices such that each edge of the graph is incident to at least one vertex in the set.

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 2 / 19

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#VertexCover

Definition

A vertex cover of a graph is a set of vertices such that each edge of the graph is incident to at least one vertex in the set.

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 2 / 19

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#VertexCover

Definition

A vertex cover of a graph is a set of vertices such that each edge of the graph is incident to at least one vertex in the set.

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 2 / 19

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#VertexCover

Definition

A vertex cover of a graph is a set of vertices such that each edge of the graph is incident to at least one vertex in the set.

X

  • Tyson Williams (UW-M)

Gadgets and Anti-Gadgets MTD 11-13-11 2 / 19

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Systematic Approach to #VertexCover

G = (V, E)

  • (u,v)∈E

OR(σ(u), σ(v)) = 1 · 1 · 1 · 1 · 1 · 1 = 1

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 3 / 19

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Systematic Approach to #VertexCover

G = (V, E)

  • (u,v)∈E

OR(σ(u), σ(v)) = 1 · 1 · 1 · 1 · 1 · 1 = 1

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 3 / 19

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Systematic Approach to #VertexCover

G = (V, E) σ : V → {0, 1}

1 1 1

  • (u,v)∈E

OR(σ(u), σ(v)) = 1 · 1 · 1 · 1 · 1 · 1 = 1

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 3 / 19

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Systematic Approach to #VertexCover

G = (V, E) σ : V → {0, 1}

1 1 1

OR OR OR OR OR OR

  • (u,v)∈E

OR(σ(u), σ(v)) = 1 · 1 · 1 · 1 · 1 · 1 = 1

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 3 / 19

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Systematic Approach to #VertexCover

G = (V, E) σ : V → {0, 1}

1 1 1

OR OR OR OR OR OR

  • (u,v)∈E

OR(σ(u), σ(v)) = 1 · 1 · 1 · 1 · 1 · 1 = 1

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 3 / 19

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Systematic Approach to #VertexCover

G = (V, E) σ : V → {0, 1}

1 1

OR OR OR OR OR OR

  • (u,v)∈E

OR(σ(u), σ(v)) = 1 · 1 · 0 · 1 · 1 · 1 = 0

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 3 / 19

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Systematic Approach to #VertexCover

G = (V, E) σ : V → {0, 1} #VertexCover(G) =

  • σ:V →{0,1}
  • (u,v)∈E

OR(σ(u), σ(v))

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 3 / 19

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Generalize

  • σ:V →{0,1}
  • (u,v)∈E

OR (σ(u), σ(v))

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 4 / 19

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Generalize

  • σ:V →{0,1}
  • (u,v)∈E

OR (σ(u), σ(v)) Input Output p q OR(p, q) 1 1 1 1 1 1 1

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 4 / 19

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Generalize

  • σ:V →{0,1}
  • (u,v)∈E

f(σ(u), σ(v)) Input Output p q OR(p, q) 1 1 1 1 1 1 1 Input Output p q f(p, q) w 1 x 1 y 1 1 z where w, x, y, z ∈ C

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 4 / 19

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Generalize

Partition Function: Z(·) Z(G) =

  • σ:V →{0,1}
  • (u,v)∈E

f(σ(u), σ(v)) Input Output p q OR(p, q) 1 1 1 1 1 1 1 Input Output p q f(p, q) w 1 x 1 y 1 1 z where w, x, y, z ∈ C

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 4 / 19

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Main Result

Theorem (Dichotomy Theorem)

Over 3-regular graphs G, the counting problem for any (binary) complex-weighted function f Z(G) =

  • σ:V →{0,1}
  • (u,v)∈E

f(σ(u), σ(v)) is either computable in polynomial time or #P-hard.

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 5 / 19

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Main Result

Theorem (Dichotomy Theorem)

Over 3-regular graphs G, the counting problem for any (binary) complex-weighted function f Z(G) =

  • σ:V →{0,1}
  • (u,v)∈E

f(σ(u), σ(v)) is either computable in polynomial time or #P-hard. Furthermore, the complexity is efficiently decidable.

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 5 / 19

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Outline

1 Related work 2 Define Holant function 3 Proof sketch

Anti-gadgets

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 6 / 19

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Related Work: Dichotomy Theorems

Symmetric f

f(0, 1) = f(1, 0)

3-regular graphs with outputs in

{0, 1} [Cai, Lu, Xia 08] {0, 1, −1} [Kowalczyk 09] R [Cai, Lu, Xia 09] C [Cai, Kowalczyk 10]

k-regular graphs with outputs in

R [Cai, Kowalczyk 10] C [Cai, Kowalczyk 11]

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 7 / 19

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Related Work: Dichotomy Theorems

Symmetric f

f(0, 1) = f(1, 0)

3-regular graphs with outputs in

{0, 1} [Cai, Lu, Xia 08] {0, 1, −1} [Kowalczyk 09] R [Cai, Lu, Xia 09] C [Cai, Kowalczyk 10]

k-regular graphs with outputs in

R [Cai, Kowalczyk 10] C [Cai, Kowalczyk 11]

This work: Asymmetric f 3-regular graphs with outputs in

C

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 7 / 19

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Definition of Holant Function

Partition Function

f f f f f f

  • σ:V →{0,1}
  • (u,v)∈E

f (σ(u), σ(v))

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 8 / 19

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Definition of Holant Function

Partition Function

Assignments to vertices Functions on edges

f f f f f f

  • σ:V →{0,1}
  • (u,v)∈E

f (σ(u), σ(v))

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 8 / 19

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Definition of Holant Function

Partition Function

Assignments to vertices Functions on edges

f f f f f f

  • σ:V →{0,1}
  • (u,v)∈E

f (σ(u), σ(v)) Holant Function

Assignment to edges Functions on vertices

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 8 / 19

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Definition of Holant Function

Partition Function

Assignments to vertices Functions on edges

f f f f f f

  • σ:V →{0,1}
  • (u,v)∈E

f (σ(u), σ(v)) Holant Function

Assignment to edges Functions on vertices

=3 =3 =3 =3 f f f f f f

  • σ:E→{0,1}
  • v∈V

gv

  • σ |E(v)
  • Tyson Williams (UW-M)

Gadgets and Anti-Gadgets MTD 11-13-11 8 / 19

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Definition of Holant Function

Holant({f}|{=3}) is a counting problem defined

  • ver (2,3)-regular bipartite

graphs. Holant Function

Assignment to edges Functions on vertices

=3 =3 =3 =3 f f f f f f

  • σ:E→{0,1}
  • v∈V

gv

  • σ |E(v)
  • Tyson Williams (UW-M)

Gadgets and Anti-Gadgets MTD 11-13-11 8 / 19

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Definition of Holant Function

Holant({f}|{=3}) is a counting problem defined

  • ver (2,3)-regular bipartite

graphs. Degree 2 vertices take f. Degree 3 vertices take =3. Holant Function

Assignment to edges Functions on vertices

=3 =3 =3 =3 f f f f f f

  • σ:E→{0,1}
  • v∈V

gv

  • σ |E(v)
  • Tyson Williams (UW-M)

Gadgets and Anti-Gadgets MTD 11-13-11 8 / 19

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Example Holant Problems

Holant({OR2}|{=3}) is #VertexCover on 3-regular graphs.

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 9 / 19

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Example Holant Problems

Holant({OR2}|{=3}) is #VertexCover on 3-regular graphs. Holant({NAND2}|{=3}) is #IndependentSet on 3-regular graphs.

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 9 / 19

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Example Holant Problems

Holant({OR2}|{=3}) is #VertexCover on 3-regular graphs. Holant({NAND2}|{=3}) is #IndependentSet on 3-regular graphs. Holant({=2}|{AT-MOST-ONE}) is #Matching.

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 9 / 19

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Example Holant Problems

Holant({OR2}|{=3}) is #VertexCover on 3-regular graphs. Holant({NAND2}|{=3}) is #IndependentSet on 3-regular graphs. Holant({=2}|{AT-MOST-ONE}) is #Matching. Holant({=2}|{EXACTLY-ONE}) is #PerfectMatching.

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 9 / 19

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General Bipartite Holant Definition

More generally, Holant(G |R) is a counting problem defined over bipartite graphs. g1 g2 g3 g4 r1 r2 r3

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 10 / 19

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General Bipartite Holant Definition

More generally, Holant(G |R) is a counting problem defined over bipartite graphs. g1 g2 g3 g4 r1 r2 r3

  • σ:E→{0,1}
  • v∈V

fv

  • σ |E(v)
  • Tyson Williams (UW-M)

Gadgets and Anti-Gadgets MTD 11-13-11 10 / 19

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Symmetric vs Asymmetric Function

=3 =3 =3 =3 f f f f f f

Input Output p q f(p, q) w 1 x 1 y 1 1 z

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 11 / 19

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Symmetric vs Asymmetric Function

=3 =3 =3 =3 f f f f f f

Input Output p q f(p, q) w 1 x 1 y 1 1 z

=3 =3 =3 =3

f f f f f f

Define p to be on the tail Define q to be on the head

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 11 / 19

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Symmetric vs Asymmetric Function

(2,3)-regular

=3 =3 =3 =3 f f f f f f

Input Output p q f(p, q) w 1 x 1 y 1 1 z Directed 3-regular

=3 =3 =3 =3

f f f f f f

Define p to be on the tail Define q to be on the head

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 11 / 19

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Strategy for Proving #P-hardness

#VertexCover is #P-hard over 3-regular graphs. Holant({OR2}|{=3}) is #VertexCover on 3-regular graphs.

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 12 / 19

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Strategy for Proving #P-hardness

#VertexCover is #P-hard over 3-regular graphs. Holant({OR2}|{=3}) is #VertexCover on 3-regular graphs. Our problem is Holant({f}|{=3}). Goal: simulate OR2 using f.

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 12 / 19

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Strategy for Proving #P-hardness

#VertexCover is #P-hard over 3-regular graphs. Holant({OR2}|{=3}) is #VertexCover on 3-regular graphs. Our problem is Holant({f}|{=3}). Goal: simulate OR2 using f. First step: Holant({OR2}|{=3}) ≤P

m Holant({f} ∪ U |{=3})

where U is the set of all unary functions.

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 12 / 19

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Strategy for Proving #P-hardness

#VertexCover is #P-hard over 3-regular graphs. Holant({OR2}|{=3}) is #VertexCover on 3-regular graphs. Our problem is Holant({f}|{=3}). Goal: simulate OR2 using f. First step: Holant({OR2}|{=3}) ≤P

m Holant({f} ∪ U |{=3})

where U is the set of all unary functions. Second step: Holant({f} ∪ U |{=3}) ≤P

T Holant({f}|{=3})

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 12 / 19

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Strategy for Proving #P-hardness

#VertexCover is #P-hard over 3-regular graphs. Holant({OR2}|{=3}) is #VertexCover on 3-regular graphs. Our problem is Holant({f}|{=3}). Goal: simulate OR2 using f. First step: Holant({OR2}|{=3}) ≤P

m Holant({f} ∪ U |{=3})

where U is the set of all unary functions. Second step: Holant({f} ∪ U |{=3}) ≤P

T Holant({f}|{=3})

Obtain U via interpolation.

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 12 / 19

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Interpolation

A degree n polynomial is uniquely defined by

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 13 / 19

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Interpolation

A degree n polynomial is uniquely defined by

n + 1 coefficients

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 13 / 19

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Interpolation

A degree n polynomial is uniquely defined by

n + 1 coefficients, or evaluations at n + 1 (different) points.

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 13 / 19

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Interpolation

A degree n polynomial is uniquely defined by

n + 1 coefficients, or evaluations at n + 1 (different) points.

Interpolation is the process of converting from evaluations to coefficients.

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 13 / 19

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Interpolation

A degree n polynomial is uniquely defined by

n + 1 coefficients, or evaluations at n + 1 (different) points.

Interpolation is the process of converting from evaluations to coefficients. We construct unary functions gi such that the evaluation points are

gi(0) gi(1).

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 13 / 19

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Interpolation

A degree n polynomial is uniquely defined by

n + 1 coefficients, or evaluations at n + 1 (different) points.

Interpolation is the process of converting from evaluations to coefficients. We construct unary functions gi such that the evaluation points are

gi(0) gi(1).

Distinct evaluation points ⇐ ⇒ unary functions pairwise linearly independent (as length-2 vectors).

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 13 / 19

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Construction of Unary Functions

Projective Gadget Recursive Gadget Unary Function

. . . . . .

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 14 / 19

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Matrix Representation

Left side indexes the row. Right side indexes the column. High order bit on top.

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 15 / 19

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Matrix Representation

Left side indexes the row. Right side indexes the column. High order bit on top. w x y z ⊗2     w x y z    

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 15 / 19

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Matrix Representation

Left side indexes the row. Right side indexes the column. High order bit on top. w x y z ⊗2     w x y z    

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 15 / 19

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Matrix Representation

Left side indexes the row. Right side indexes the column. High order bit on top. w x y z ⊗2     w x y z     w x y z ⊗2     w y x z    

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 15 / 19

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Matrix Representation

Left side indexes the row. Right side indexes the column. High order bit on top. w x y z ⊗2     w x y z     w x y z ⊗2     w y x z     Matrix of the composition is the product of the component matrices.

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 15 / 19

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Anti-Gadget Construction

Want set of matrix powers to form an infinite set of pairwise linearly independent matrices.

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 16 / 19

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Anti-Gadget Construction

Want set of matrix powers to form an infinite set of pairwise linearly independent matrices. If this matrix has this property, then we are done. w x y z ⊗2     w x y z    

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 16 / 19

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Anti-Gadget Construction

Want set of matrix powers to form an infinite set of pairwise linearly independent matrices. If this matrix has this property, then we are done. w x y z ⊗2     w x y z     Otherwise, some power k is a multiple of the identity matrix.

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 16 / 19

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Anti-Gadget Construction

Want set of matrix powers to form an infinite set of pairwise linearly independent matrices. If this matrix has this property, then we are done. w x y z ⊗2     w x y z     Otherwise, some power k is a multiple of the identity matrix. Using only k − 1 compositions creates an anti-gadget.

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 16 / 19

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Anti-Gadget Construction

Want set of matrix powers to form an infinite set of pairwise linearly independent matrices. If this matrix has this property, then we are done. w x y z ⊗2     w x y z     Otherwise, some power k is a multiple of the identity matrix. Using only k − 1 compositions creates an anti-gadget.     w x y z ⊗2     w x y z        

−1

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 16 / 19

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Anti-Gadget Construction

Want set of matrix powers to form an infinite set of pairwise linearly independent matrices. If this matrix has this property, then we are done. w x y z ⊗2     w x y z     Otherwise, some power k is a multiple of the identity matrix. Using only k − 1 compositions creates an anti-gadget.         w x y z        

−1 w

x y z ⊗2−1

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 16 / 19

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Anti-Gadget Technique

        w x y z        

−1 w

x y z ⊗2−1

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 17 / 19

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Anti-Gadget Technique

        w x y z        

−1 w

x y z ⊗2−1 w x y z ⊗2     w y x z    

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 17 / 19

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Anti-Gadget Technique

        w x y z        

−1 w

x y z ⊗2−1 w x y z ⊗2     w y x z     The composition of these two gadgets yields...

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 17 / 19

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Anti-Gadget Technique

        w x y z        

−1 w

x y z ⊗2−1 w x y z ⊗2     w y x z     The composition of these two gadgets yields...     1

y x x y

1    

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 17 / 19

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The First Anti-Gadget Lemma

Lemma

For w, x, y, z ∈ C, if wz = xy, wxyz = 0, and |x| = |y|, then there exists a recursive gadget whose matrix powers form an infinite set of pairwise linearly independent matrices.

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 18 / 19

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The First Anti-Gadget Lemma

Lemma

For w, x, y, z ∈ C, if wz = xy, wxyz = 0, and |x| = |y|, then there exists a recursive gadget whose matrix powers form an infinite set of pairwise linearly independent matrices.

Corollary

For w, x, y, z ∈ C as above, Holant({f}|{=3}) is #P-hard.

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 18 / 19

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Thank You

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 19 / 19

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Thank You

Paper and slides available on my website. www.cs.wisc.edu/~tdw

Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 19 / 19