Edge Coloring,Siegel's Theorem,and a HolantDichotomy Jin-Yi Cai, Heng Guo , Tyson Williams University of Wisconsin-Madison Beijing Sep 11th 2014 Heng Guo (CS, UW-Madison) Edge Colorings China Theory Week 2014 1 / 25
Edge Coloring Definition Heng Guo (CS, UW-Madison) Edge Colorings China Theory Week 2014 2 / 25
Obvious lower bound is G . Given G , deciding if G colors suffice is NP -complete over 3-regular graphs [ Holyer 81 ], k -regular graphs for k 3 [ Leven, Galil 83 ]. Optimization version for multi-graphs. Deciding Edge Colorings Theorem (Vizing's Theorem) Edge coloring using at most ∆ ( G ) + 1 colors exists in simple graphs. Heng Guo (CS, UW-Madison) Edge Colorings China Theory Week 2014 3 / 25
Given G , deciding if G colors suffice is NP -complete over 3-regular graphs [ Holyer 81 ], k -regular graphs for k 3 [ Leven, Galil 83 ]. Optimization version for multi-graphs. Deciding Edge Colorings Theorem (Vizing's Theorem) Edge coloring using at most ∆ ( G ) + 1 colors exists in simple graphs. Obvious lower bound is ∆ ( G ) . Heng Guo (CS, UW-Madison) Edge Colorings China Theory Week 2014 3 / 25
Optimization version for multi-graphs. Deciding Edge Colorings Theorem (Vizing's Theorem) Edge coloring using at most ∆ ( G ) + 1 colors exists in simple graphs. Obvious lower bound is ∆ ( G ) . Given G , deciding if ∆ ( G ) colors suffice is NP -complete over 3-regular graphs [ Holyer 81 ], k -regular graphs for k ⩾ 3 [ Leven, Galil 83 ]. Heng Guo (CS, UW-Madison) Edge Colorings China Theory Week 2014 3 / 25
Deciding Edge Colorings Theorem (Vizing's Theorem) Edge coloring using at most ∆ ( G ) + 1 colors exists in simple graphs. Obvious lower bound is ∆ ( G ) . Given G , deciding if ∆ ( G ) colors suffice is NP -complete over 3-regular graphs [ Holyer 81 ], k -regular graphs for k ⩾ 3 [ Leven, Galil 83 ]. Optimization version for multi-graphs. Heng Guo (CS, UW-Madison) Edge Colorings China Theory Week 2014 3 / 25
Theorem # - EdgeColoring is # P -hard over r -regular planar graphs for all r 3. No edge colorings if r . (There is no regular planar graph of degree larger than or equal to 6 by counting the average degree of a triangulation. Our result is actually for multi-graphs when r 5.) Counting Edge Colorings Problem : # κ - EdgeColoring . Input : A graph G . Output : Number of edge colorings of G using at most κ colors. Heng Guo (CS, UW-Madison) Edge Colorings China Theory Week 2014 4 / 25
No edge colorings if r . (There is no regular planar graph of degree larger than or equal to 6 by counting the average degree of a triangulation. Our result is actually for multi-graphs when r 5.) Counting Edge Colorings Problem : # κ - EdgeColoring . Input : A graph G . Output : Number of edge colorings of G using at most κ colors. Theorem # κ - EdgeColoring is # P -hard over r -regular planar graphs for all κ ⩾ r ⩾ 3. Heng Guo (CS, UW-Madison) Edge Colorings China Theory Week 2014 4 / 25
(There is no regular planar graph of degree larger than or equal to 6 by counting the average degree of a triangulation. Our result is actually for multi-graphs when r 5.) Counting Edge Colorings Problem : # κ - EdgeColoring . Input : A graph G . Output : Number of edge colorings of G using at most κ colors. Theorem # κ - EdgeColoring is # P -hard over r -regular planar graphs for all κ ⩾ r ⩾ 3. No edge colorings if κ < r . Heng Guo (CS, UW-Madison) Edge Colorings China Theory Week 2014 4 / 25
Counting Edge Colorings Problem : # κ - EdgeColoring . Input : A graph G . Output : Number of edge colorings of G using at most κ colors. Theorem # κ - EdgeColoring is # P -hard over r -regular planar graphs for all κ ⩾ r ⩾ 3. No edge colorings if κ < r . (There is no regular planar graph of degree larger than or equal to 6 by counting the average degree of a triangulation. Our result is actually for multi-graphs when r > 5.) Heng Guo (CS, UW-Madison) Edge Colorings China Theory Week 2014 4 / 25
To compute # - EdgeColoring , we sum over all configurations: Holant G AD 3 w E G A configuration is a proper coloring if and only if it satisfies all constraints, that is, w AD 3 1 E v v V G Counting Edge Colorings as a Holant Problem AD 3 AD 3 Put the local constraint function AD 3 on each node. AD 3 AD 3 { 1 if x , y , z ∈ [ κ ] are distinct AD 3 ( x , y , z ) = 0 otherwise. AD 3 AD 3 Heng Guo (CS, UW-Madison) Edge Colorings China Theory Week 2014 5 / 25
To compute # - EdgeColoring , we sum over all configurations: Holant G AD 3 w E G Counting Edge Colorings as a Holant Problem AD 3 AD 3 Put the local constraint function AD 3 on each node. AD 3 AD 3 { 1 if x , y , z ∈ [ κ ] are distinct AD 3 ( x , y , z ) = 0 otherwise. AD 3 AD 3 A configuration is a proper coloring if and only if it satisfies all constraints, that is, ∏ w ( σ ) = AD 3 ( ) = 1 . σ | E ( v ) v ∈ V ( G ) Heng Guo (CS, UW-Madison) Edge Colorings China Theory Week 2014 5 / 25
To compute # - EdgeColoring , we sum over all configurations: Holant G AD 3 w E G Counting Edge Colorings as a Holant Problem AD 3 AD 3 Put the local constraint function AD 3 on each node. AD 3 AD 3 { 1 if x , y , z ∈ [ κ ] are distinct AD 3 ( x , y , z ) = 0 otherwise. AD 3 AD 3 A configuration is a proper coloring if and only if it satisfies all constraints, that is, ∏ w ( σ ) = AD 3 ( ) = 1 . σ | E ( v ) v ∈ V ( G ) Heng Guo (CS, UW-Madison) Edge Colorings China Theory Week 2014 5 / 25
Counting Edge Colorings as a Holant Problem AD 3 AD 3 Put the local constraint function AD 3 on each node. AD 3 AD 3 { 1 if x , y , z ∈ [ κ ] are distinct AD 3 ( x , y , z ) = 0 otherwise. AD 3 AD 3 A configuration is a proper coloring if and only if it satisfies all constraints, that is, ∏ w ( σ ) = AD 3 ( ) = 1 . σ | E ( v ) v ∈ V ( G ) To compute # κ - EdgeColoring , we sum over all configurations: ∑ Holant ( G ; AD 3 ) = w ( σ ) . σ : E ( G ) → [ κ ] Heng Guo (CS, UW-Madison) Edge Colorings China Theory Week 2014 5 / 25
The Holant problem is to compute Holant G f f E v E G v V G aka: tensor network contraction, factor graphs, … Holant Problems In this talk, we consider all local constraint functions a if x = y = z (all equal) f ( x , y , z ) = b otherwise c if x ̸ = y ̸ = z ̸ = x (all distinct). Denote f by ⟨ a , b , c ⟩ . Then AD 3 = ⟨ 0 , 0 , 1 ⟩ . Heng Guo (CS, UW-Madison) Edge Colorings China Theory Week 2014 6 / 25
aka: tensor network contraction, factor graphs, … Holant Problems In this talk, we consider all local constraint functions a if x = y = z (all equal) f ( x , y , z ) = b otherwise c if x ̸ = y ̸ = z ̸ = x (all distinct). Denote f by ⟨ a , b , c ⟩ . Then AD 3 = ⟨ 0 , 0 , 1 ⟩ . The Holant problem is to compute ∑ ∏ Holant κ ( G ; f ) = f ( ) σ | E ( v ) . σ : E ( G ) → [ κ ] v ∈ V ( G ) Heng Guo (CS, UW-Madison) Edge Colorings China Theory Week 2014 6 / 25
Holant Problems In this talk, we consider all local constraint functions a if x = y = z (all equal) f ( x , y , z ) = b otherwise c if x ̸ = y ̸ = z ̸ = x (all distinct). Denote f by ⟨ a , b , c ⟩ . Then AD 3 = ⟨ 0 , 0 , 1 ⟩ . The Holant problem is to compute ∑ ∏ Holant κ ( G ; f ) = f ( ) σ | E ( v ) . σ : E ( G ) → [ κ ] v ∈ V ( G ) aka: tensor network contraction, factor graphs, … Heng Guo (CS, UW-Madison) Edge Colorings China Theory Week 2014 6 / 25
# - EdgeColoring in 3-regular graphs is the special case a b c 0 0 1 , and it is # P -hard. Main Theorem Theorem For any domain size κ ⩾ 3 and any a , b , c ∈ C , the problem of computing Holant κ (− ; ⟨ a , b , c ⟩ ) is either # P -hard or in polynomial time, even when the input is restricted to planar graphs. Heng Guo (CS, UW-Madison) Edge Colorings China Theory Week 2014 7 / 25
Main Theorem Theorem For any domain size κ ⩾ 3 and any a , b , c ∈ C , the problem of computing Holant κ (− ; ⟨ a , b , c ⟩ ) is either # P -hard or in polynomial time, even when the input is restricted to planar graphs. # κ - EdgeColoring in 3-regular graphs is the special case ⟨ a , b , c ⟩ = ⟨ 0 , 0 , 1 ⟩ , and it is # P -hard. Heng Guo (CS, UW-Madison) Edge Colorings China Theory Week 2014 7 / 25
Trivial problems: 0 0 0 , 1 1 1 . 1 Up to a holographic transformation, it is one of the following: 2 The support of each constraints contains at most many pair-wise disjoint assignments, such as equalities. Solvable by Gaussian sums. Some tractable cases are not so obvious, for example, 3 and Holant 3 5 2 4 ; 4 and Holant 4 3 4 i 1 1 2 i . We have a simple procedure to verify. Tractable Problems Tractable problems are: Heng Guo (CS, UW-Madison) Edge Colorings China Theory Week 2014 8 / 25
Up to a holographic transformation, it is one of the following: 2 The support of each constraints contains at most many pair-wise disjoint assignments, such as equalities. Solvable by Gaussian sums. Some tractable cases are not so obvious, for example, 3 and Holant 3 5 2 4 ; 4 and Holant 4 3 4 i 1 1 2 i . We have a simple procedure to verify. Tractable Problems Tractable problems are: Trivial problems: ⟨ 0 , 0 , 0 ⟩ , ⟨ 1 , 1 , 1 ⟩ . 1 Heng Guo (CS, UW-Madison) Edge Colorings China Theory Week 2014 8 / 25
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