A Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai 1 , - - PowerPoint PPT Presentation

▶

a holant dichotomy is the fkt algorithm universal

A Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai 1 , - - PowerPoint PPT Presentation

Dec 29, 2022 157 likes •736 views

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

slide-1

SLIDE 1

A Holant Dichotomy: Is the FKT Algorithm Universal?

Jin-Yi Cai1, Zhiguo Fu2, Heng Guo1, and Tyson Williams 1

1University of Wisconsin-Madison 2Jilin University

Berkeley, CA Oct 20, 2015

Heng Guo (UW-Madison) Planar Holant FOCS 2015 1 / 20

slide-2

SLIDE 2

Ising Model

Edge interaction

[

β 1 1 β

]

1 β β 1 1 β β 1 1 β β 1 β 1 1 β

Heng Guo (UW-Madison) Planar Holant FOCS 2015 2 / 20

slide-3

SLIDE 3

Ising Model

Edge interaction

[

β 1 1 β

]

1 β β 1 1 β β 1 1 β β 1 β 1 1 β

Configuration σ : V → {0, 1} w(σ) = β8 Pr(σ) ∼ w(σ)

Heng Guo (UW-Madison) Planar Holant FOCS 2015 2 / 20

slide-4

SLIDE 4

Ising Model

Edge interaction

[

β 1 1 β

]

1 β β 1 1 β β 1 1 β β 1 β 1 1 β

Configuration σ : V → {0, 1} w(σ) = β0 = 1 Pr(σ) ∼ w(σ)

Heng Guo (UW-Madison) Planar Holant FOCS 2015 2 / 20

slide-5

SLIDE 5

Ising Model

Edge interaction

[

β 1 1 β

]

1 β β 1 1 β β 1 1 β β 1 β 1 1 β

Configuration σ : V → {0, 1} w(σ) = β4 Pr(σ) ∼ w(σ)

Heng Guo (UW-Madison) Planar Holant FOCS 2015 2 / 20

slide-6

SLIDE 6

Ising Model

Edge interaction

[

β 1 1 β

]

1 β β 1 1 β β 1 1 β β 1 β 1 1 β

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

slide-7

SLIDE 7

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

slide-8

SLIDE 8

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

slide-9

SLIDE 9

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

slide-10

SLIDE 10

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

slide-11

SLIDE 11

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

slide-12

SLIDE 12

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

slide-13

SLIDE 13

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

slide-14

SLIDE 14

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

slide-15

SLIDE 15

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

slide-16

SLIDE 16

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

slide-17

SLIDE 17

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

slide-18

SLIDE 18

#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

slide-19

SLIDE 19

#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

slide-20

SLIDE 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

slide-21

SLIDE 21

#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

slide-22

SLIDE 22

Counting Perfect Matchings

Perfect Matchings

f1 f2 f1 f3 f1 f3 f4 f2 Heng Guo (UW-Madison) Planar Holant FOCS 2015 6 / 20

slide-23

SLIDE 23

Counting Perfect Matchings

Perfect Matchings

f1 f2 f1 f3 f1 f3 f4 f2 Heng Guo (UW-Madison) Planar Holant FOCS 2015 6 / 20

slide-24

SLIDE 24

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

slide-25

SLIDE 25

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

slide-26

SLIDE 26

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

slide-27

SLIDE 27

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

slide-28

SLIDE 28

#PM as a Holant

Put functions EXACTONE (EO) on nodes (edges are variables).

Heng Guo (UW-Madison) Planar Holant FOCS 2015 9 / 20

slide-29

SLIDE 29

#PM as a Holant

Put functions EXACTONE (EO) on nodes (edges are variables).

EO3 EO4 EO3 EO4 EO3 EO4 EO3 EO4 Heng Guo (UW-Madison) Planar Holant FOCS 2015 9 / 20

slide-30

SLIDE 30

#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 EO3 EO4 EO3 EO4 Heng Guo (UW-Madison) Planar Holant FOCS 2015 9 / 20

slide-31

SLIDE 31

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

slide-32

SLIDE 32

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

slide-33

SLIDE 33

Main Result

Let F be a set of symmetric complex-weighted Boolean functions. Pl-Holant(F) is #P-hard unless

Heng Guo (UW-Madison) Planar Holant FOCS 2015 11 / 20

slide-34

SLIDE 34

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;

Heng Guo (UW-Madison) Planar Holant FOCS 2015 11 / 20

slide-35

SLIDE 35

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,

Heng Guo (UW-Madison) Planar Holant FOCS 2015 11 / 20

slide-36

SLIDE 36

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.

Heng Guo (UW-Madison) Planar Holant FOCS 2015 11 / 20

slide-37

SLIDE 37

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].

Heng Guo (UW-Madison) Planar Holant FOCS 2015 11 / 20

slide-38

SLIDE 38

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

slide-39

SLIDE 39

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

slide-40

SLIDE 40

#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.

Heng Guo (UW-Madison) Planar Holant FOCS 2015 13 / 20

slide-41

SLIDE 41

#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

slide-42

SLIDE 42

The Algorithm

The algorithm is based on recursively simplifying the instance, until it can be solved by known algorithms such as FKT.

Heng Guo (UW-Madison) Planar Holant FOCS 2015 14 / 20

slide-43

SLIDE 43

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.

Heng Guo (UW-Madison) Planar Holant FOCS 2015 14 / 20

slide-44

SLIDE 44

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.

Heng Guo (UW-Madison) Planar Holant FOCS 2015 14 / 20

slide-45

SLIDE 45

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

slide-46

SLIDE 46

Examplary Planar Structures

Lemma

Let 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

slide-47

SLIDE 47

A Score Based Proof

Assign a score sv to each vertex v ∈ V so that ∑

v∈V

sv = |V| − |E| + |F| = 2 > 0.

Heng Guo (UW-Madison) Planar Holant FOCS 2015 16 / 20

slide-48

SLIDE 48

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|: 1

k each;

▶ −|E|: − 7

12 for degree 3 and − 5 12 for the other, or − 1 2 each.

Heng Guo (UW-Madison) Planar Holant FOCS 2015 16 / 20

slide-49

SLIDE 49

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|: 1

k each;

▶ −|E|: − 7

12 for degree 3 and − 5 12 for the other, or − 1 2 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

slide-50

SLIDE 50

Proof Roadmap of the Main Theorem

Identification

f 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

slide-51

SLIDE 51

Guidance or Misguidance?

We start the whole project with the belief that HA with matchgates captures all planar tractable cases . . .

Heng Guo (UW-Madison) Planar Holant FOCS 2015 18 / 20

slide-52

SLIDE 52

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].)

Heng Guo (UW-Madison) Planar Holant FOCS 2015 18 / 20

slide-53

SLIDE 53

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).

Heng Guo (UW-Madison) Planar Holant FOCS 2015 18 / 20

slide-54

SLIDE 54

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

slide-55

SLIDE 55

Concluding Remarks

A sharp algebraic separation exists between tractable and #P-hard problems.

Heng Guo (UW-Madison) Planar Holant FOCS 2015 19 / 20

slide-56

SLIDE 56

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

slide-57

SLIDE 57

Thank You!

Heng Guo (UW-Madison) Planar Holant FOCS 2015 20 / 20