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
# 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
# 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
# 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
# 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
Systematic Approach to # VertexCover G = ( V, E ) � OR( σ ( u ) , σ ( v )) = 1 · 1 · 1 · 1 · 1 · 1 = 1 ( u,v ) ∈ E Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 3 / 19
Systematic Approach to # VertexCover G = ( V, E ) � OR( σ ( u ) , σ ( v )) = 1 · 1 · 1 · 1 · 1 · 1 = 1 ( u,v ) ∈ E Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 3 / 19
Systematic Approach to # VertexCover 0 G = ( V, E ) σ : V → { 0 , 1 } 1 1 1 � OR( σ ( u ) , σ ( v )) = 1 · 1 · 1 · 1 · 1 · 1 = 1 ( u,v ) ∈ E Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 3 / 19
Systematic Approach to # VertexCover 0 G = ( V, E ) σ : V → { 0 , 1 } OR OR OR 1 OR OR 1 1 OR � OR( σ ( u ) , σ ( v )) = 1 · 1 · 1 · 1 · 1 · 1 = 1 ( u,v ) ∈ E Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 3 / 19
Systematic Approach to # VertexCover 0 G = ( V, E ) σ : V → { 0 , 1 } OR OR OR 1 OR OR 1 1 OR � OR( σ ( u ) , σ ( v )) = 1 · 1 · 1 · 1 · 1 · 1 = 1 ( u,v ) ∈ E Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 3 / 19
Systematic Approach to # VertexCover 0 G = ( V, E ) σ : V → { 0 , 1 } OR OR OR 1 OR OR 1 0 OR � OR( σ ( u ) , σ ( v )) = 1 · 1 · 0 · 1 · 1 · 1 = 0 ( u,v ) ∈ E Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 3 / 19
Systematic Approach to # VertexCover G = ( V, E ) σ : V → { 0 , 1 } � � # VertexCover ( G ) = OR( σ ( u ) , σ ( v )) σ : V →{ 0 , 1 } ( u,v ) ∈ E Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 3 / 19
Generalize � � OR ( σ ( u ) , σ ( v )) σ : V →{ 0 , 1 } ( u,v ) ∈ E Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 4 / 19
Generalize � � OR ( σ ( u ) , σ ( v )) σ : V →{ 0 , 1 } ( u,v ) ∈ E Input Output OR( p, q ) p q 0 0 0 0 1 1 1 0 1 1 1 1 Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 4 / 19
Generalize � � f ( σ ( u ) , σ ( v )) σ : V →{ 0 , 1 } ( u,v ) ∈ E Input Output Input Output OR( p, q ) f ( p, q ) p q p q 0 0 0 0 0 w 0 1 1 0 1 x 1 0 1 1 0 y 1 1 1 1 1 z where w, x, y, z ∈ C Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 4 / 19
Generalize Partition Function: Z ( · ) � � Z ( G ) = f ( σ ( u ) , σ ( v )) σ : V →{ 0 , 1 } ( u,v ) ∈ E Input Output Input Output OR( p, q ) f ( p, q ) p q p q 0 0 0 0 0 w 0 1 1 0 1 x 1 0 1 1 0 y 1 1 1 1 1 z where w, x, y, z ∈ C Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 4 / 19
Main Result Theorem (Dichotomy Theorem) Over 3-regular graphs G , the counting problem for any (binary) complex-weighted function f � � Z ( G ) = f ( σ ( u ) , σ ( v )) σ : V →{ 0 , 1 } ( u,v ) ∈ E is either computable in polynomial time or #P -hard. Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 5 / 19
Main Result Theorem (Dichotomy Theorem) Over 3-regular graphs G , the counting problem for any (binary) complex-weighted function f � � Z ( G ) = f ( σ ( u ) , σ ( v )) σ : V →{ 0 , 1 } ( u,v ) ∈ E 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
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
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
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
Definition of Holant Function Partition Function f f f f f f � � f ( σ ( u ) , σ ( v )) σ : V →{ 0 , 1 } ( u,v ) ∈ E Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 8 / 19
Definition of Holant Function Partition Function Assignments to vertices Functions on edges f f f f f f � � f ( σ ( u ) , σ ( v )) σ : V →{ 0 , 1 } ( u,v ) ∈ E Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 8 / 19
Definition of Holant Function Partition Function Holant Function Assignments to vertices Assignment to edges Functions on edges Functions on vertices f f f f f f � � f ( σ ( u ) , σ ( v )) σ : V →{ 0 , 1 } ( u,v ) ∈ E Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 8 / 19
Definition of Holant Function Partition Function Holant Function Assignments to vertices Assignment to edges Functions on edges Functions on vertices = 3 f f f f f f = 3 f f f f f f = 3 = 3 � � � � � � σ | E ( v ) g v f ( σ ( u ) , σ ( v )) σ : E →{ 0 , 1 } v ∈ V σ : V →{ 0 , 1 } ( u,v ) ∈ E Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 8 / 19
Definition of Holant Function Holant Function Holant( { f }|{ = 3 } ) is a counting problem defined Assignment to edges Functions on vertices over (2,3)-regular bipartite graphs. = 3 f f f = 3 f f f = 3 = 3 � � � � σ | E ( v ) g v σ : E →{ 0 , 1 } v ∈ V Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 8 / 19
Definition of Holant Function Holant Function Holant( { f }|{ = 3 } ) is a counting problem defined Assignment to edges Functions on vertices over (2,3)-regular bipartite graphs. = 3 Degree 2 vertices take f . Degree 3 vertices take = 3 . f f f = 3 f f f = 3 = 3 � � � � σ | E ( v ) g v σ : E →{ 0 , 1 } v ∈ V Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 8 / 19
Example Holant Problems Holant( { OR 2 }|{ = 3 } ) is # VertexCover on 3-regular graphs. Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 9 / 19
Example Holant Problems Holant( { OR 2 }|{ = 3 } ) is # VertexCover on 3-regular graphs. Holant( { NAND 2 }|{ = 3 } ) is # IndependentSet on 3-regular graphs. Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 9 / 19
Example Holant Problems Holant( { OR 2 }|{ = 3 } ) is # VertexCover on 3-regular graphs. Holant( { NAND 2 }|{ = 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
Example Holant Problems Holant( { OR 2 }|{ = 3 } ) is # VertexCover on 3-regular graphs. Holant( { NAND 2 }|{ = 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
General Bipartite Holant Definition More generally, Holant( G |R ) is a counting problem defined over bipartite graphs. g 1 r 1 g 2 r 2 g 3 r 3 g 4 Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 10 / 19
General Bipartite Holant Definition More generally, Holant( G |R ) is a counting problem defined over bipartite graphs. g 1 r 1 g 2 r 2 g 3 r 3 g 4 � � � � f v σ | E ( v ) v ∈ V σ : E →{ 0 , 1 } Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 10 / 19
Symmetric vs Asymmetric Function Input Output f ( p, q ) p q = 3 0 0 w 0 1 x f 1 0 y 1 1 z f f = 3 f f f = 3 = 3 Tyson Williams (UW-M) Gadgets and Anti-Gadgets MTD 11-13-11 11 / 19
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