A Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai 1 , Zhiguo Fu 2 , Heng Guo 1 , and Tyson Williams 1 1 University of Wisconsin-Madison 2 Jilin University Berkeley, CA Oct 20, 2015 Heng Guo (UW-Madison) Planar Holant FOCS 2015 1 / 20
Ising Model [ ] β 1 Edge interaction 1 β β β 1 1 β 1 β 1 β β 1 1 1 β β 1 Heng Guo (UW-Madison) Planar Holant FOCS 2015 2 / 20
Ising Model [ ] β 1 Edge interaction 1 β β β 1 1 β 1 β 1 β β 1 1 1 β β 1 Configuration σ : V → { 0, 1 } w ( σ ) = β 8 Pr ( σ ) ∼ w ( σ ) Heng Guo (UW-Madison) Planar Holant FOCS 2015 2 / 20
Ising Model [ ] β 1 Edge interaction 1 β β β 1 1 β 1 β 1 β β 1 1 1 β β 1 Configuration σ : V → { 0, 1 } w ( σ ) = β 0 = 1 Pr ( σ ) ∼ w ( σ ) Heng Guo (UW-Madison) Planar Holant FOCS 2015 2 / 20
Ising Model [ ] β 1 Edge interaction 1 β β β 1 1 β 1 β 1 β β 1 1 1 β β 1 Configuration σ : V → { 0, 1 } w ( σ ) = β 4 Pr ( σ ) ∼ w ( σ ) Heng Guo (UW-Madison) Planar Holant FOCS 2015 2 / 20
Ising Model [ ] β 1 Edge interaction 1 β β β 1 1 β 1 1 β β β 1 1 1 β β 1 Partition function (normalizing factor): ∑ Z G ( β ) = w ( σ ) σ : V → { 0,1 } where w ( σ ) = β m ( σ ) , m ( σ ) is the number of monochromatic edges under σ . Heng Guo (UW-Madison) Planar Holant FOCS 2015 2 / 20
FKT Algorithm Computing the partition function of the Ising model is # P -hard unless in some degenerate cases. Heng Guo (UW-Madison) Planar Holant FOCS 2015 3 / 20
FKT Algorithm Computing the partition function of the Ising model is # P -hard unless in some degenerate cases. For planar graphs, there is a polynomial time algorithm [Kastelyn 61 & 67, Temperley and Fisher 61]. Heng Guo (UW-Madison) Planar Holant FOCS 2015 3 / 20
FKT Algorithm Computing the partition function of the Ising model is # P -hard unless in some degenerate cases. For planar graphs, there is a polynomial time algorithm [Kastelyn 61 & 67, Temperley and Fisher 61]. Reduction to #PM (counting perfect matchings) in planar graphs. Heng Guo (UW-Madison) Planar Holant FOCS 2015 3 / 20
FKT Algorithm Computing the partition function of the Ising model is # P -hard unless in some degenerate cases. For planar graphs, there is a polynomial time algorithm [Kastelyn 61 & 67, Temperley and Fisher 61]. Reduction to #PM (counting perfect matchings) in planar graphs. ▶ #PM is # P -hard [Valiant 79] in general graphs as well. Heng Guo (UW-Madison) Planar Holant FOCS 2015 3 / 20
FKT Algorithm Computing the partition function of the Ising model is # P -hard unless in some degenerate cases. For planar graphs, there is a polynomial time algorithm [Kastelyn 61 & 67, Temperley and Fisher 61]. Reduction to #PM (counting perfect matchings) in planar graphs. ▶ #PM is # P -hard [Valiant 79] in general graphs as well. #PM can be computed via Pfaffian orientations of planar graphs. Heng Guo (UW-Madison) Planar Holant FOCS 2015 3 / 20
Holographic Algorithms Valiant introduced holographic algorithms to extend the reach of FKT algorithms [Valiant 04]: Heng Guo (UW-Madison) Planar Holant FOCS 2015 4 / 20
Holographic Algorithms Valiant introduced holographic algorithms to extend the reach of FKT algorithms [Valiant 04]: Matchgates: functions expressible by perfect matchings. Heng Guo (UW-Madison) Planar Holant FOCS 2015 4 / 20
Holographic Algorithms Valiant introduced holographic algorithms to extend the reach of FKT algorithms [Valiant 04]: Matchgates: functions expressible by perfect matchings. Holographic Transformation: a change of basis. Heng Guo (UW-Madison) Planar Holant FOCS 2015 4 / 20
Holographic Algorithms Valiant introduced holographic algorithms to extend the reach of FKT algorithms [Valiant 04]: Matchgates: functions expressible by perfect matchings. Holographic Transformation: a change of basis. A series of work (see e.g. [Cai and Lu 07]) characterizes what problems can be solved by holographic algorithms based on matchgates. Heng Guo (UW-Madison) Planar Holant FOCS 2015 4 / 20
Holographic Algorithms Valiant introduced holographic algorithms to extend the reach of FKT algorithms [Valiant 04]: Matchgates: functions expressible by perfect matchings. Holographic Transformation: a change of basis. A series of work (see e.g. [Cai and Lu 07]) characterizes what problems can be solved by holographic algorithms based on matchgates. It still leaves open the question of whether holographic algorithms solve # P -hard problems? Heng Guo (UW-Madison) Planar Holant FOCS 2015 4 / 20
Holographic Algorithms Valiant introduced holographic algorithms to extend the reach of FKT algorithms [Valiant 04]: Matchgates: functions expressible by perfect matchings. Holographic Transformation: a change of basis. A series of work (see e.g. [Cai and Lu 07]) characterizes what problems can be solved by holographic algorithms based on matchgates. It still leaves open the question of whether holographic algorithms solve # P -hard problems? We need to answer this question in some framework. Heng Guo (UW-Madison) Planar Holant FOCS 2015 4 / 20
#CSP A natural generalization of the Ising partition function is Counting Constraint Satisfaction Problems (with weights). Heng Guo (UW-Madison) Planar Holant FOCS 2015 5 / 20
#CSP A natural generalization of the Ising partition function is Counting Constraint Satisfaction Problems (with weights). ▶ Vertex-coloring model — vertices are variables and edges are functions. Heng Guo (UW-Madison) Planar Holant FOCS 2015 5 / 20
#CSP A natural generalization of the Ising partition function is Counting Constraint Satisfaction Problems (with weights). ▶ Vertex-coloring model — vertices are variables and edges are functions. ▶ Edges (pairwise) → hyperedges (multi-party). Heng Guo (UW-Madison) Planar Holant FOCS 2015 5 / 20
#CSP A natural generalization of the Ising partition function is Counting Constraint Satisfaction Problems (with weights). ▶ Vertex-coloring model — vertices are variables and edges are functions. ▶ Edges (pairwise) → hyperedges (multi-party). Name #CSP ( F ) Instance A bipartite graph G = ( V , C , E ) and a mapping π : C → F Output The quantity: ∑ ∏ ( ) f c σ | N ( c ) , σ : V → { 0,1 } c ∈ C where N ( c ) are the neighbors of c and f c = π ( c ) ∈ F . Heng Guo (UW-Madison) Planar Holant FOCS 2015 5 / 20
Counting Perfect Matchings Perfect Matchings f 1 f 2 f 1 f 2 f 3 f 4 f 3 f 1 Heng Guo (UW-Madison) Planar Holant FOCS 2015 6 / 20
Counting Perfect Matchings Perfect Matchings f 1 f 2 f 1 f 2 f 3 f 4 f 3 f 1 Heng Guo (UW-Madison) Planar Holant FOCS 2015 6 / 20
Holant Problems #PM is provably not expressible in vertex assignment models. (see e.g. [Freedman, Lovász, and Schrijver 07]) Heng Guo (UW-Madison) Planar Holant FOCS 2015 7 / 20
Holant Problems #PM is provably not expressible in vertex assignment models. (see e.g. [Freedman, Lovász, and Schrijver 07]) Edge-coloring models — edges are variables and vertices are functions. Heng Guo (UW-Madison) Planar Holant FOCS 2015 7 / 20
Holant Problems #PM is provably not expressible in vertex assignment models. (see e.g. [Freedman, Lovász, and Schrijver 07]) Edge-coloring models — edges are variables and vertices are functions. Name Holant ( F ) Instance A graph G = ( V , E ) and a mapping π : V → F Output The quantity: ∑ ∏ ( ) σ | E ( v ) f v , v ∈ V σ : E → { 0,1 } where E ( v ) are the incident edges of v and f v = π ( v ) ∈ F . Heng Guo (UW-Madison) Planar Holant FOCS 2015 7 / 20
More general than #CSP: #CSP ( F ) ≡ T Holant ( EQ ∪ F ) , where EQ = { = 1 , = 2 , = 3 , . . . } is the set of equalities of all arities. Equivalent formulation: Tensor network contraction . . . Pl-Holant ( F ) denotes the version where instances are all planar. Heng Guo (UW-Madison) Planar Holant FOCS 2015 8 / 20
#PM as a Holant Put functions E XACT O NE (EO) on nodes (edges are variables). Heng Guo (UW-Madison) Planar Holant FOCS 2015 9 / 20
#PM as a Holant Put functions E XACT O NE (EO) on nodes (edges are variables). EO 3 EO 4 EO 3 EO 4 EO 4 EO 3 EO 4 EO 3 Heng Guo (UW-Madison) Planar Holant FOCS 2015 9 / 20
#PM as a Holant Put functions E XACT O NE (EO) on nodes (edges are variables). #PM is then the partition function: ∑ ∏ #PM = EO d ( σ | E ( v ) ) . σ : E → { 0,1 } v ∈ V EO 3 EO 4 EO 3 EO 4 EO 4 EO 3 EO 4 EO 3 Heng Guo (UW-Madison) Planar Holant FOCS 2015 9 / 20
Complexity Classifications Counting problems with local constraints are usually classified into: 1. P -time solvable over general graphs; 2. # P -hard over general graphs but P -time solvable over planar graphs; 3. # P -hard over planar graphs. Heng Guo (UW-Madison) Planar Holant FOCS 2015 10 / 20
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