![Page 1: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/1.jpg)
A Holant Dichotomy:
Is the FKT Algorithm Universal?
Jin-Yi Cai1, Zhiguo Fu2, Heng Guo1, and Tyson Williams 1
1University of Wisconsin-Madison2Jilin University
Berkeley, CA
Oct 20, 2015
Heng Guo (UW-Madison) Planar Holant FOCS 2015 1 / 20
![Page 2: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/2.jpg)
Ising Model
Edge interaction[β 11 β
]1β
β1
1β
β1
1ββ1
β11β
Partition function (normalizing factor):
ZG(β) =∑
σ:V→{0,1}
w(σ)
where w(σ) = βm(σ), m(σ) is the number of monochromatic edges under σ.
Heng Guo (UW-Madison) Planar Holant FOCS 2015 2 / 20
![Page 3: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/3.jpg)
Ising Model
Edge interaction[β 11 β
]1β
β1
1β
β1
1ββ1
β11β
Configuration σ : V → {0, 1}
w(σ) = β8
Pr(σ) ∼ w(σ)
Partition function (normalizing factor):
ZG(β) =∑
σ:V→{0,1}
w(σ)
where w(σ) = βm(σ), m(σ) is the number of monochromatic edges under σ.
Heng Guo (UW-Madison) Planar Holant FOCS 2015 2 / 20
![Page 4: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/4.jpg)
Ising Model
Edge interaction[β 11 β
]1β
β1
1β
β1
1ββ1
β11β
Configuration σ : V → {0, 1}
w(σ) = β0 = 1
Pr(σ) ∼ w(σ)
Partition function (normalizing factor):
ZG(β) =∑
σ:V→{0,1}
w(σ)
where w(σ) = βm(σ), m(σ) is the number of monochromatic edges under σ.
Heng Guo (UW-Madison) Planar Holant FOCS 2015 2 / 20
![Page 5: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/5.jpg)
Ising Model
Edge interaction[β 11 β
]1β
β1
1β
β1
1ββ1
β11β
Configuration σ : V → {0, 1}
w(σ) = β4
Pr(σ) ∼ w(σ)
Partition function (normalizing factor):
ZG(β) =∑
σ:V→{0,1}
w(σ)
where w(σ) = βm(σ), m(σ) is the number of monochromatic edges under σ.
Heng Guo (UW-Madison) Planar Holant FOCS 2015 2 / 20
![Page 6: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/6.jpg)
Ising Model
Edge interaction[β 11 β
]1β
β1
1β
β1
1ββ1
β11β
Partition function (normalizing factor):
ZG(β) =∑
σ:V→{0,1}
w(σ)
where w(σ) = βm(σ), m(σ) is the number of monochromatic edges under σ.
Heng Guo (UW-Madison) Planar Holant FOCS 2015 2 / 20
![Page 7: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/7.jpg)
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
![Page 8: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/8.jpg)
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
![Page 9: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/9.jpg)
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
![Page 10: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/10.jpg)
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
![Page 11: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/11.jpg)
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
![Page 12: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/12.jpg)
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
![Page 13: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/13.jpg)
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
![Page 14: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/14.jpg)
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
![Page 15: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/15.jpg)
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
![Page 16: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/16.jpg)
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
![Page 17: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/17.jpg)
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
![Page 18: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/18.jpg)
#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: ∑σ:V→{0,1}
∏c∈C
fc(σ |N(c)
),
where N(c) are the neighbors of c and fc = π(c) ∈ F.
Heng Guo (UW-Madison) Planar Holant FOCS 2015 5 / 20
![Page 19: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/19.jpg)
#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: ∑σ:V→{0,1}
∏c∈C
fc(σ |N(c)
),
where N(c) are the neighbors of c and fc = π(c) ∈ F.
Heng Guo (UW-Madison) Planar Holant FOCS 2015 5 / 20
![Page 20: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/20.jpg)
#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: ∑σ:V→{0,1}
∏c∈C
fc(σ |N(c)
),
where N(c) are the neighbors of c and fc = π(c) ∈ F.
Heng Guo (UW-Madison) Planar Holant FOCS 2015 5 / 20
![Page 21: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/21.jpg)
#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: ∑σ:V→{0,1}
∏c∈C
fc(σ |N(c)
),
where N(c) are the neighbors of c and fc = π(c) ∈ F.
Heng Guo (UW-Madison) Planar Holant FOCS 2015 5 / 20
![Page 22: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/22.jpg)
Counting Perfect Matchings
Perfect Matchings
f1 f2 f1
f3
f1f3f4
f2
Heng Guo (UW-Madison) Planar Holant FOCS 2015 6 / 20
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Counting Perfect Matchings
Perfect Matchings
f1 f2 f1
f3
f1f3f4
f2
Heng Guo (UW-Madison) Planar Holant FOCS 2015 6 / 20
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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→{0,1}
∏v∈V
fv(σ |E(v)
),
where E(v) are the incident edges of v and fv = π(v) ∈ F.
Heng Guo (UW-Madison) Planar Holant FOCS 2015 7 / 20
![Page 25: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/25.jpg)
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→{0,1}
∏v∈V
fv(σ |E(v)
),
where E(v) are the incident edges of v and fv = π(v) ∈ F.
Heng Guo (UW-Madison) Planar Holant FOCS 2015 7 / 20
![Page 26: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/26.jpg)
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→{0,1}
∏v∈V
fv(σ |E(v)
),
where E(v) are the incident edges of v and fv = π(v) ∈ F.
Heng Guo (UW-Madison) Planar Holant FOCS 2015 7 / 20
![Page 27: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/27.jpg)
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
![Page 28: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/28.jpg)
#PM as a Holant
Put functions EXACTONE (EO) on nodes (edges are variables).
#PM is then the partition function:
#PM =∑
σ:E→{0,1}
∏v∈V
EOd(σ |E(v)).
EO3 EO4 EO3
EO4
EO3EO4EO3
EO3
Heng Guo (UW-Madison) Planar Holant FOCS 2015 9 / 20
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#PM as a Holant
Put functions EXACTONE (EO) on nodes (edges are variables).
#PM is then the partition function:
#PM =∑
σ:E→{0,1}
∏v∈V
EOd(σ |E(v)).
EO3 EO4 EO3
EO4
EO3EO4EO3
EO4
Heng Guo (UW-Madison) Planar Holant FOCS 2015 9 / 20
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#PM as a Holant
Put functions EXACTONE (EO) on nodes (edges are variables).
#PM is then the partition function:
#PM =∑
σ:E→{0,1}
∏v∈V
EOd(σ |E(v)).
EO3 EO4 EO3
EO4
EO3EO4EO3
EO4
Heng Guo (UW-Madison) Planar Holant FOCS 2015 9 / 20
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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.
Category (2) is always captured by holographic algorithms with matchgates.
Examples include:
Tutte polynomials [Vertigan 91], [Vertigan 05].
Spin systems [Kowalczyk 10], [Cai, Kowalczyk, Williams 12].
#CSP [Cai, Lu, Xia 10], [G. and Williams 13].
Heng Guo (UW-Madison) Planar Holant FOCS 2015 10 / 20
![Page 32: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/32.jpg)
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.
Category (2) is always captured by holographic algorithms with matchgates.
Examples include:
Tutte polynomials [Vertigan 91], [Vertigan 05].
Spin systems [Kowalczyk 10], [Cai, Kowalczyk, Williams 12].
#CSP [Cai, Lu, Xia 10], [G. and Williams 13].
Heng Guo (UW-Madison) Planar Holant FOCS 2015 10 / 20
![Page 33: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/33.jpg)
Main Result
Let F be a set of symmetric complex-weighted Boolean functions.
Pl-Holant(F) is #P-hard unless
1. Holant(F) is tractable;
2. there exists a holographic transformation under which F is matchgate,
3. F defines a special class of problems to count orientations.
Category (1) is characterized in [Cai, G., Williams 13].
Category (3) is not captured by holographic algorithms with matchgates!
Heng Guo (UW-Madison) Planar Holant FOCS 2015 11 / 20
![Page 34: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/34.jpg)
Main Result
Let F be a set of symmetric complex-weighted Boolean functions.
Pl-Holant(F) is #P-hard unless
1. Holant(F) is tractable;
2. there exists a holographic transformation under which F is matchgate,
3. F defines a special class of problems to count orientations.
Category (1) is characterized in [Cai, G., Williams 13].
Category (3) is not captured by holographic algorithms with matchgates!
Heng Guo (UW-Madison) Planar Holant FOCS 2015 11 / 20
![Page 35: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/35.jpg)
Main Result
Let F be a set of symmetric complex-weighted Boolean functions.
Pl-Holant(F) is #P-hard unless
1. Holant(F) is tractable;
2. there exists a holographic transformation under which F is matchgate,
3. F defines a special class of problems to count orientations.
Category (1) is characterized in [Cai, G., Williams 13].
Category (3) is not captured by holographic algorithms with matchgates!
Heng Guo (UW-Madison) Planar Holant FOCS 2015 11 / 20
![Page 36: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/36.jpg)
Main Result
Let F be a set of symmetric complex-weighted Boolean functions.
Pl-Holant(F) is #P-hard unless
1. Holant(F) is tractable;
2. there exists a holographic transformation under which F is matchgate,
3. F defines a special class of problems to count orientations.
Category (1) is characterized in [Cai, G., Williams 13].
Category (3) is not captured by holographic algorithms with matchgates!
Heng Guo (UW-Madison) Planar Holant FOCS 2015 11 / 20
![Page 37: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/37.jpg)
Main Result
Let F be a set of symmetric complex-weighted Boolean functions.
Pl-Holant(F) is #P-hard unless
1. Holant(F) is tractable;
2. there exists a holographic transformation under which F is matchgate,
3. F defines a special class of problems to count orientations.
Category (1) is characterized in [Cai, G., Williams 13].
Category (3) is not captured by holographic algorithms with matchgates!
Heng Guo (UW-Madison) Planar Holant FOCS 2015 11 / 20
![Page 38: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/38.jpg)
Main Result
Let F be a set of symmetric complex-weighted Boolean functions.
Pl-Holant(F) is #P-hard unless
1. Holant(F) is tractable;
2. there exists a holographic transformation under which F is matchgate,
3. F defines a special class of problems to count orientations.
Category (1) is characterized in [Cai, G., Williams 13].
Category (3) is not captured by holographic algorithms with matchgates!
Heng Guo (UW-Madison) Planar Holant FOCS 2015 11 / 20
![Page 39: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/39.jpg)
New Planar Tractable Case
Counting Orientations, where two types of nodes are allowed:
1. Exactly one edge coming in;
2. All edges coming in or going out (either a sink or a source).
Moreover, we require that the gcd of the degrees of type 2 nodes is at
least 5.
Then the problem is tractable.
Heng Guo (UW-Madison) Planar Holant FOCS 2015 12 / 20
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#PM in Planar Hypergraphs
As a special case of our result, consider the following problem.
Name #Planar-Hyper-PM(S)
Instance A hypergraph H whose incidence graph is planar, and
hyperedge sizes are prescribed by S.
Output The number of perfect matchings in H.
Let t = gcd(S).
If t ⩾ 5 or S ⊆ {1, 2},
then #Planar-Hyper-PM(S) is computable in polynomial time.
Otherwise t ⩽ 4, S ̸⊆ {1, 2}, and #Planar-Hyper-PM(S) is #P-hard.
Heng Guo (UW-Madison) Planar Holant FOCS 2015 13 / 20
![Page 41: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/41.jpg)
#PM in Planar Hypergraphs
As a special case of our result, consider the following problem.
Name #Planar-Hyper-PM(S)
Instance A hypergraph H whose incidence graph is planar, and
hyperedge sizes are prescribed by S.
Output The number of perfect matchings in H.
Let t = gcd(S).
If t ⩾ 5 or S ⊆ {1, 2},
then #Planar-Hyper-PM(S) is computable in polynomial time.
Otherwise t ⩽ 4, S ̸⊆ {1, 2}, and #Planar-Hyper-PM(S) is #P-hard.
Heng Guo (UW-Madison) Planar Holant FOCS 2015 13 / 20
![Page 42: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/42.jpg)
The Algorithm
The algorithm is based on recursively simplifying the instance, until it
can be solved by known algorithms such as FKT.
The planar constraint guarantees the existence of certain structures
that can be simplified.
Some steps of the process may provide orientations inconsistent with
the original instance, but we can keep track of enough information to
go back and check.
Tractable mainly due to degree rigidity.
Heng Guo (UW-Madison) Planar Holant FOCS 2015 14 / 20
![Page 43: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/43.jpg)
The Algorithm
The algorithm is based on recursively simplifying the instance, until it
can be solved by known algorithms such as FKT.
The planar constraint guarantees the existence of certain structures
that can be simplified.
Some steps of the process may provide orientations inconsistent with
the original instance, but we can keep track of enough information to
go back and check.
Tractable mainly due to degree rigidity.
Heng Guo (UW-Madison) Planar Holant FOCS 2015 14 / 20
![Page 44: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/44.jpg)
The Algorithm
The algorithm is based on recursively simplifying the instance, until it
can be solved by known algorithms such as FKT.
The planar constraint guarantees the existence of certain structures
that can be simplified.
Some steps of the process may provide orientations inconsistent with
the original instance, but we can keep track of enough information to
go back and check.
Tractable mainly due to degree rigidity.
Heng Guo (UW-Madison) Planar Holant FOCS 2015 14 / 20
![Page 45: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/45.jpg)
The Algorithm
The algorithm is based on recursively simplifying the instance, until it
can be solved by known algorithms such as FKT.
The planar constraint guarantees the existence of certain structures
that can be simplified.
Some steps of the process may provide orientations inconsistent with
the original instance, but we can keep track of enough information to
go back and check.
Tractable mainly due to degree rigidity.
Heng Guo (UW-Madison) Planar Holant FOCS 2015 14 / 20
![Page 46: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/46.jpg)
Examplary Planar Structures
LemmaLet G = (L ∪ R, E) be a planar bipartite graph with parts L and R.
Every vertex in L has degree at least 5;
every vertex in R has degree at least 3.
If G is simple, then there exists one of the two wheel structures in G.
· · ·
· · ·· · ·
(a) Type 1
· · · · · ·
· · ·· · ·
(b) Type 2
Heng Guo (UW-Madison) Planar Holant FOCS 2015 15 / 20
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A Score Based Proof
Assign a score sv to each vertex v ∈ V so that∑v∈V
sv = |V |− |E |+ |F | = 2 > 0.
▶ |V |: +1 each;
▶ |F |: 1k each;
▶ −|E |: − 712 for degree 3 and − 5
12 for the other, or − 12 each.
If no wheel structure exists, then there exists a 1-1 mapping between
positive vertices and negative vertices, and negative scores are larger.
Hence the total score has to be negative. Contradiction.
Heng Guo (UW-Madison) Planar Holant FOCS 2015 16 / 20
![Page 48: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/48.jpg)
A Score Based Proof
Assign a score sv to each vertex v ∈ V so that∑v∈V
sv = |V |− |E |+ |F | = 2 > 0.
▶ |V |: +1 each;
▶ |F |: 1k each;
▶ −|E |: − 712 for degree 3 and − 5
12 for the other, or − 12 each.
If no wheel structure exists, then there exists a 1-1 mapping between
positive vertices and negative vertices, and negative scores are larger.
Hence the total score has to be negative. Contradiction.
Heng Guo (UW-Madison) Planar Holant FOCS 2015 16 / 20
![Page 49: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/49.jpg)
A Score Based Proof
Assign a score sv to each vertex v ∈ V so that∑v∈V
sv = |V |− |E |+ |F | = 2 > 0.
▶ |V |: +1 each;
▶ |F |: 1k each;
▶ −|E |: − 712 for degree 3 and − 5
12 for the other, or − 12 each.
If no wheel structure exists, then there exists a 1-1 mapping between
positive vertices and negative vertices, and negative scores are larger.
Hence the total score has to be negative. Contradiction.
Heng Guo (UW-Madison) Planar Holant FOCS 2015 16 / 20
![Page 50: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/50.jpg)
Proof Roadmap of the Main Theorem
Identification
of tractable
signatures
Previous
dichotomy
theorems
New
hardness
proofs
Single
signature
dichotomy
Mixing
New
tractable
problems
Pl-#CSP2
dichotomy
p. 63—p. 128
Final
dichotomy
Heng Guo (UW-Madison) Planar Holant FOCS 2015 17 / 20
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Guidance or Misguidance?
We start the whole project with the belief that HA with matchgates captures
all planar tractable cases . . .
. . . until we were stuck proving Pl-#CSPd dichotomies.
(The non-planar version is an important stepping stone in previous work
[Huang and Lu 12] and [Cai, G., Williams 13].)
The natural generalization for d ⩾ 5 does not hold, and in the end we proved
the d = 2 case (where the natural generalization does hold).
However lots of progress was made due to this belief.
Heng Guo (UW-Madison) Planar Holant FOCS 2015 18 / 20
![Page 52: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/52.jpg)
Guidance or Misguidance?
We start the whole project with the belief that HA with matchgates captures
all planar tractable cases . . .
. . . until we were stuck proving Pl-#CSPd dichotomies.
(The non-planar version is an important stepping stone in previous work
[Huang and Lu 12] and [Cai, G., Williams 13].)
The natural generalization for d ⩾ 5 does not hold, and in the end we proved
the d = 2 case (where the natural generalization does hold).
However lots of progress was made due to this belief.
Heng Guo (UW-Madison) Planar Holant FOCS 2015 18 / 20
![Page 53: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/53.jpg)
Guidance or Misguidance?
We start the whole project with the belief that HA with matchgates captures
all planar tractable cases . . .
. . . until we were stuck proving Pl-#CSPd dichotomies.
(The non-planar version is an important stepping stone in previous work
[Huang and Lu 12] and [Cai, G., Williams 13].)
The natural generalization for d ⩾ 5 does not hold, and in the end we proved
the d = 2 case (where the natural generalization does hold).
However lots of progress was made due to this belief.
Heng Guo (UW-Madison) Planar Holant FOCS 2015 18 / 20
![Page 54: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/54.jpg)
Guidance or Misguidance?
We start the whole project with the belief that HA with matchgates captures
all planar tractable cases . . .
. . . until we were stuck proving Pl-#CSPd dichotomies.
(The non-planar version is an important stepping stone in previous work
[Huang and Lu 12] and [Cai, G., Williams 13].)
The natural generalization for d ⩾ 5 does not hold, and in the end we proved
the d = 2 case (where the natural generalization does hold).
However lots of progress was made due to this belief.
Heng Guo (UW-Madison) Planar Holant FOCS 2015 18 / 20
![Page 55: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/55.jpg)
Concluding Remarks
A sharp algebraic separation exists between tractable and #P-hard
problems.
There exists planar tractable cases that are not captured by
holographic algorithms with matchages (or FKT).
Heng Guo (UW-Madison) Planar Holant FOCS 2015 19 / 20
![Page 56: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/56.jpg)
Concluding Remarks
A sharp algebraic separation exists between tractable and #P-hard
problems.
There exists planar tractable cases that are not captured by
holographic algorithms with matchages (or FKT).
Heng Guo (UW-Madison) Planar Holant FOCS 2015 19 / 20
![Page 57: A Holant Dichotomy: Is the FKT Algorithm Universal?homepages.inf.ed.ac.uk/hguo/slides/Planar.Holant.pdfA Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai1, Zhiguo Fu2,](https://reader030.vdocuments.mx/reader030/viewer/2022040210/5e5962c790a6463b9262951b/html5/thumbnails/57.jpg)
Thank You!
Heng Guo (UW-Madison) Planar Holant FOCS 2015 20 / 20