on the largest pcc graphs - math-lucaghidelli.site graphs - ottawa 2017.pdfthecombinatorialcurvature...

On the largest PCC graphs

Luca Ghidelli

2017

Plan of the talk

1 The combinatorial curvaturePolyhedra and abstract graphs

2 The PCC graphsDefinition and main questionLower bounds and upper bounds

3 Integer Linear Programming proofOldridge implementationLP, ILP, MILP: algorithms and software

4 Discharging proofTransportation planThe last two cases

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The combinatorial curvature Polyhedra and abstract graphs

Curvature for convex polyhedra

12π

∑angles = 5

6 = 0.83...

12π

∑angles = 1

12π

∑angles = 2

3 = 0.66....

12π

∑angles = 7

12 = 0.58...

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Formula for the curvature

curvature(v) := 1− 12π∑

angles

= 1− 12π∑f∼v

(π − 2π|f |

)

= 1− deg(v)2 +

∑f∼v

1|f |

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Discrete Gauss-Bonnet

• Formula for the combinatorial curvature:

curvature(v) = 1− deg(v)2 +

∑f∼v

1|f | .

• Gauss-Bonnet formula:

total curvature =∑

v

(1−

∑e∼v

12 +

∑f∼v

1|f |

)= #V −#E + #F = 2.

N.B. Euler-Poincaré characteristic: χ(sphere) = 2.

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Abstract planar graphs

∪ {∞}

8 vertices of type (4,4,4), ↑.

c(4, 4, 4) = 34

Vertices of type ↖ (3,3,3), (3,3,10), (3,3,3,10), (3,10,10).

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The PCC graphs Definition and main question

Prisms and Antiprisms

c(4, 4, 6) = 16

c(3, 3, 3, 6) = 16

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PCC graphs

DefinitionA finite graph G is a PCC graph if:

• is "planar" (embedded in sphere)• is simple (no multiple edges)• deg(v) ≥ 3, all v ∈ V• c(v) > 0, all v ∈ V• is not prism or antiprism.

Rmk: a positively curved planar graph as above cannot be infinite(DeVos-Mohar 2007).

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Finiteness

Theorem (G. 2017)G PCC, f ∈ F ⇒ |f | ≤ 41.

Corollary (Chen-Chen 2008)G PCC, v ∈ V ⇒ c(v) > 1

1722 .

Corollary (DeVos-Mohar 2007)G PCC⇒ #V ≤ 3444

ProblemMAX #Vertices, if G PCC?

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The PCC graphs Lower bounds and upper bounds

Great Rhombicosidodecahedron

#V = 120

c(4, 6, 10) = 160

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Réti-Bitai-Kosztoláni (2005)

#V = 138

c(4, 4, 19) = 119

c(4, 5, 19) = 1190

c(3, 4, 4, 5) = 130

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Nicholson-Sneddon (2011)

#V = 208

c(3, 11, 11) = 166

c(3, 11, 13) = 1858

c(3, 3, 3, 13) = 113

c(3, 3, 4, 11) = 1132

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G. (2011), Oldridge (2017)

#V = 208

c(3, 7, 39) = 1546

c(3, 3, 3, 39) = 139

c(3, 3, 5, 7) = 1105

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Upper bounds

• (DeVos-Mohar 2007) #V ≤ 3444.• (Chen-Chen 2008) #V ≤ 3444.• (Zhang 2008) #V ≤ 579, wrong.• (G. 2011) #V ≤ 364, 264, 218, (2013) #V ≤ 210.• (Byung-Geun Oh 2017) #V ≤ 380.• (Oldridge 2017) #V ≤ 244, conditional (Integer LinearProgramming).

• (G. 2017) #V ≤ 208 (Discharging).• Work in progress: combine the two last approaches.

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Combinatorial complexity

∼ 200 vertices, 365 types: (3,8,23), (3,3,5,7), (5,5,9), ...

Naive complexityA dumb computer search requires ∼ 10500 operations.

• Facts: 1010 is already a lot, and 1020 is way too much.• Moore’s law, Graphene, 3D, Optical, DNA, Quantum...• Fun facts (The Singularity is Near, Kurzweil): a universe-scaleLlyoid ultimate black-hole-storage computer achievingBeckenstein information-entropy bound would perform2.8× 10229 operations in 8.8× 10131 years.

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Integer Linear Programming proof Oldridge implementation

ILP - variables

• nV = #Vertices, nE = #Edges, nF = #Faces;

• v1, . . . , vT = {# ,# ,# , . . .};

• f3, . . . , f41 = {# ,# ,# , . . .};

• α1, . . . , αA = {# , . . .};

• β1, . . . , βB = {# ,# , . . .}/ ≈.

Number of variablesT = 345, A ≈ 1000, B ≈ 1.000.000.

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ILP - constraints

• nV =∑

t vt ;• 2nE =

∑t |t| vt ;

• nF =∑

s fs ;• 2 = nV − nE + nF ;• sfs =

∑t m(s, t)vt (39 equations);

• fs =∑|b|=s βb (39 equations);

• αa =∑

b m′(a, b)βb (≈ 1000 equations);• αa =

∑t m′(a, t)vt (≈ 1000 equations).

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ILP - outcome

The ILP optimization programGiven {VARIABLES≥ 0}

{CONSTRAINTS}

Maximize quantity nV

Output nV ≤ 264.

Rmk: a refinement of the constraints gives nV ≤ 244.

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Integer Linear Programming proof LP, ILP, MILP: algorithms and software

Linear programming

LP is one of the most successful techniques in Operations Research.

Standard form:(slack augmented)

Variables x ≥ 0Constraints Ax = bMaximize cTx

Symplex algorithm.

Complexity of LP:theoretically not clear; inpractice very good.

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Integer Linear Programming

Standard form:(slack augmented)

Variables x ∈ Zx ≥ 0

Constraints Ax = bMaximize cTx

• Complexity: ILP is NP-hard.• LP relaxation + cutting plane; branch and bound (enumerationof candidates); branch and cut.

• Many heuristic approaches.• MILP = Mixed Integer Linear Program.

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Software

• files: .lp .mp• SCIP, ZIMPL, CMPL/Colip• ...

C++ program ⇒ outputs .cmplCMPL/Colip ⇒ outputs answer

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Discharging proof Transportation plan

Discharging - the idea

• Goal: avg(c : V :→ R≥0) ≥ 0.0048 . . .But: c(3, 3, 3) = 0.5� c(3, 7, 41) = 0.00058 . . .

• Idea: modify "vertex ∈ face" (fuzzy)

φ : V × F → [0, 1].

• Transportation plan (measure theory) / Discharging(combinatorics)

c ′(f ) =∑

v φ(v , f )c(v)∑v φ(v , f )

Goal: c ′ : F → R≥0 "smooth", s.t. avg(c ′) ≈ min(c ′)

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Discharging - examples 1/2

41

v7 6

41

v φ(v) = [41]

v11 11 φ(v) = 12 [11] + 1

2 [11′]

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Discharging - examples 2/2

11

v w2w1if type w1,w2 6= (4, 5, 19)

φ(v) = [11]else φ(v) = [5]

11

13ba v

if a, b = 11φ(v) = 3

7 [11] + 47 [13]

else φ(v) = 17 [11] + 6

7 [13].

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Discharging - outcome

• Analysis of 63 cases, verification of 37 short linear inequalitiessatisfied by 32 quantities.

Outcome of case analysisFor all f ∈ F we have c ′(f ) ≥ 1

105 .

Corollary#V ≤ 210

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Discharging proof The last two cases

Case #V = 210

5 7

7 5

5

5

5 5

6 6C

7

67 7A

7

57

5

7

5B

Double-count the edges

A = 3B , edges (5,6)2A = 3C , edges (5,7)3B = 2C , edges (6,7)

Contradiction.

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Case #V = 209

• There is a chain of(3,5,7)-triangles.

• T = #triangles: multiple of 4|G1|+ |G2|+ 3

2T = 209 is odd.⇒ WLOG |G1| < |G2|

• Do graph surgery.Contradiction.

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Thank you

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on the largest pcc graphs - math-lucaghidelli.site graphs - ottawa 2017.pdfthecombinatorialcurvature...

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