Discrete Volume Computations for Polyhedramath.sfsu.edu/beck/papers/nantes.pdf · Discrete Volume Computations for Polyhedra x y z 5 20 2 Matthias Beck San Francisco State University

Discrete Volume Computations

for Polyhedra

x

y

z

5

20

2

Matthias Beck

San Francisco State University

math.sfsu.edu/beck

DGCI 2016

Digital

“Science is what we understand well enough to explain to a computer, artis all the rest.”

Donald Knuth

Discrete Volume Computations for Polyhedra Matthias Beck 2

Themes

Discrete-geometricpolynomials

Generatingfunctions

Discrete Fourieranalysis

Combinatorialstructures

Computation(complexity)

(0, 1, 2, 3)

(1, 0, 2, 3)

(1, 0, 3, 2)

(0, 1, 3, 2)

(0, 2, 1, 3)

(2, 0, 1, 3)

(2, 1, 0, 3)

(1, 2, 0, 3)

(0, 2, 3, 1)

(0, 3, 2, 1)

(0, 3, 1, 2)

(3, 0, 1, 2)

(3, 1, 0, 2)

(1, 3, 0, 2)

(1, 3, 2, 0)

(1, 2, 3, 0)

(2, 0, 3, 1)

(2, 3, 0, 1)

(2, 3, 1, 0)

(2, 1, 3, 0)

(3, 2, 0, 1)

(3, 0, 2, 1)

(3, 2, 1, 0)

(3, 1, 2, 0)


A Sample Problem: Birkhoff–von Neumann Polytope

Bn =

x11 · · · x1n

... ...xn1 . . . xnn

∈ Rn2

≥0 :

∑j xjk = 1 for all 1 ≤ k ≤ n∑k xjk = 1 for all 1 ≤ j ≤ n


Discrete Volumes

Rational polyhedron P ⊂ Rd – solution set of a system of linear equalities& inequalities with integer coefficients

Goal: understand P ∩ Zd . . .

I (list)∑

m∈P∩Zdzm1

1 zm22 · · · z

mdd

I (count)∣∣P ∩ Zd

∣∣


Discrete Volumes



I (list)∑

m∈P∩Zdzm1

1 zm22 · · · z

mdd


∣∣I (volume) vol(P) = lim

t→∞

1

td

∣∣∣∣P ∩ 1

tZd∣∣∣∣


Discrete Volumes



I (list)∑

m∈P∩Zdzm1

1 zm22 · · · z

mdd


∣∣I (volume) vol(P) = lim

t→∞

1

td

∣∣∣∣P ∩ 1

tZd∣∣∣∣

Ehrhart function LP(t) :=

∣∣∣∣P ∩ 1

tZd∣∣∣∣ =

∣∣tP ∩ Zd∣∣ for t ∈ Z>0


Some Motivation

I Linear systems are everywhere, and so polyhedra are everywhere.


Some Motivation


I In applications, the volume of the polytope represented by a linearsystem measures some fundamental data of this system (“average”).


Some Motivation



I Polytopes are basic geometric objects, yet even for these basic objectsvolume computation is hard and there remain many open problems.


Some Motivation



I Polytopes are basic geometric objects, yet even for these basic objectsvolume computation is hard and there remain many open problems.

I Many discrete problems in various mathematical areas are linearproblems, thus they ask for the discrete volume of a polytope indisguise.


A Warm-Up Ehrhart Function

Lattice polytope P ⊂ Rd – convex hull of finitely points in Zd

For t ∈ Z>0 let LP(t) := #(tP ∩ Zd

)Example:

∆ = conv {(0, 0), (1, 0), (0, 1)}

={

(x, y) ∈ R2≥0 : x+ y ≤ 1

}





)Example:

∆ = conv {(0, 0), (1, 0), (0, 1)}

={

(x, y) ∈ R2≥0 : x+ y ≤ 1

}

L∆(t) =

(t+ 2

2

)= 1

2(t+ 1)(t+ 2)





)Example:

∆ = conv {(0, 0), (1, 0), (0, 1)}

={

(x, y) ∈ R2≥0 : x+ y ≤ 1

}

L∆(t) =

(t+ 2

2

)= 1

2(t+ 1)(t+ 2) ,

a polynomial in t with leading coefficient vol (∆) =1

2





)Example:

∆ = conv {(0, 0), (1, 0), (0, 1)}

={

(x, y) ∈ R2≥0 : x+ y ≤ 1

}

L∆(t) =

(t+ 2

2

)= 1

2(t+ 1)(t+ 2)

comes with the friendly generating function∑t≥0

(t+ 2

2

)zt =

1

(1− z)3


Ehrhart Polynomials

Theorem (Ehrhart 1962) For any lattice polytope P,LP(t) is a polynomial in t of degree dimP withleading coefficient volP and constant term 1.

Equivalently, EhrP(z) := 1 +∑t≥1

LP(t) zt is rational:

EhrP(z) =h(z)

(1− z)dimP+1

where the Ehrhart h-vector h(z) satisfies h(0) = 1 andh(1) = (dimP)! vol(P).


Ehrhart Polynomials

Theorem (Ehrhart 1962) For any lattice polytope P,LP(t) is a polynomial in t of degree dimP withleading coefficient volP and constant term 1.

Equivalently, EhrP(z) := 1 +∑t≥1

LP(t) zt is rational:

EhrP(z) =h(z)

(1− z)dimP+1

where the Ehrhart h-vector h(z) satisfies h(0) = 1 andh(1) = (dimP)! vol(P).

Seeming dichotomy: vol(P) = limt→∞

1

tdimPLP(t) can be computed discretely

via a finite amount of data.


Ehrhart Polynomials

Theorem (Ehrhart 1962) For any lattice polytope P,LP(t) is a polynomial in t of degree d := dimP withleading coefficient volP and constant term 1.

EhrP(z) := 1 +∑t≥1

LP(t) zt =h(z)

(1− z)d+1

Equivalent descriptions of an Ehrhart polynomial:

I LP(t) = cd td + cd−1 t

d−1 + · · ·+ c0

I via roots of LP(t)

I EhrP(z) −→ LP(t) = h0

(t+dd

)+ h1

(t+d−1d

)+ · · ·+ hd

(td

)


Ehrhart Polynomials


EhrP(z) := 1 +∑t≥1

LP(t) zt =h(z)

(1− z)d+1

−→ LP(t) = h0

(t+dd

)+ h1

(t+d−1d

)+ · · ·+ hd

(td

)Theorem (Macdonald 1971) (−1)dLP(−t) enumerates the interior latticepoints in tP. Equivalently,

LP◦(t) = hd(t+d−1d

)+ hd−1

(t+d−2d

)+ · · ·+ h0

(t−1d

)


Ehrhart Polynomials


EhrP(z) := 1 +∑t≥1

LP(t) zt =h(z)

(1− z)d+1

−→ LP◦(t) = hd(t+d−1d

)+ hd−1

(t+d−2d

)+ · · ·+ h0

(t−1d

)Theorem (Stanley 1980) h0, h1, . . . , hd are nonnegative integers.


Ehrhart Polynomials


EhrP(z) := 1 +∑t≥1

LP(t) zt =h(z)

(1− z)d+1

−→ LP◦(t) = hd(t+d−1d

)+ hd−1

(t+d−2d

)+ · · ·+ h0

(t−1d

)Theorem (Stanley 1980) h0, h1, . . . , hd are nonnegative integers.

Corollary If hd+1−k > 0 then kP◦ contains an integer point.


Interlude: Graph Coloring a la Ehrhart

χK2(k) = 2

(k

2

)...

1k +

1k +

1 = x 2x

2K

(Blass–Sagan)


Interlude: Graph Coloring a la Ehrhart

χK2(k) = 2

(k

2

)...

1k +

1k +

1 = x 2x

2K

(Blass–Sagan)Similarly, for any given graphG on n nodes, we can write

χG(k) = a0

(k + n

n

)+ a1

(k + n− 1

n

)+ · · ·+ an

(k

n

)for some (meaningful) nonnegative integers a0, . . . , an


Computational Complexity of Integer-Point Transforms


−→ σP(z) :=∑

m∈P∩Zdzm1

1 zm22 · · · z

mdd is a rational function in z1, z2, . . . , zd

Lenstra (1983) polynomial-time algorithm to decide whether σP(z) = 0

Barvinok (1994) polynomial-time algorithm to compute σP(z)




−→ σP(z) :=∑

m∈P∩Zdzm1

1 zm22 · · · z




Example P = [0, 1000]

σ[0,1000](z) = 1 + z + · · ·+ z1000 =1− z1001

1− z




−→ σP(z) :=∑

m∈P∩Zdzm1

1 zm22 · · · z




Given a polytope P we can compute

EhrP(z) = σcone(P)(1, 1, . . . , 1, z)

where cone(P) := R≥0 (P × {1})




−→ σP(z) :=∑

m∈P∩Zdzm1

1 zm22 · · · z




Implementations:

De Loera, Koppe et al www.math.ucdavis.edu/∼latte

Verdoolaege freshmeat.net/projects/barvinok


Ehrhart Quasipolynomials

Rational polytope P ⊂ Rd – convex hull of finitely points in Qd

Theorem (Ehrhart 1962) LP(t) is a quasipolynomial in t:

LP(t) = cd(t) td + cd−1(t) td−1 + · · ·+ c0(t)

where c0(t), . . . , cd(t) are periodic functions.

Example L[0,12](t) =1

2t+

3 + (−1)t

4


Ehrhart Quasipolynomials


Theorem (Ehrhart 1962) LP(t) is a quasipolynomial in t:

LP(t) = cd(t) td + cd−1(t) td−1 + · · ·+ c0(t)

where c0(t), . . . , cd(t) are periodic functions. Equivalently,

EhrP(z) := 1 +∑t≥1

LP(t) zt =h(z)

(1− zp)dimP+1

for some (minimal) p ∈ Z>0 (the period of LP(t)).

Example Ehr[0,12](z) =∑t≥0

(1

2t+

3 + (−1)t

4

)zt =

1 + z

(1− z2)2


Interlude: Magic Squares joint withThomas ZaslavskyAndrew Van Herick

Mn(t) – number of n× n squares with distinctentries and row, column, and diagonal sums t

4 9 23 5 78 1 6

Similar to graph coloring, this is a polyhedral

problem with forbidden hyperplanes:

inside-out polytopes


Interlude: Magic Squaresjoint withThomas ZaslavskyAndrew Van Herick

Mn(t) – number of n× n squares with distinctentries and row, column, and diagonal sums t

4 9 23 5 78 1 6

Example

M3(t) =

2t2−32t+1449 = 2

9(t2 − 16t+ 72) if t ≡ 0 mod 182t2−32t+78

9 = 29(t− 3)(t− 13) if t ≡ 3 mod 18

2t2−32t+1209 = 2

9(t− 6)(t− 10) if t ≡ 6 mod 182t2−32t+126

9 = 29(t− 7)(t− 9) if t ≡ 9 mod 18

2t2−32t+969 = 2

9(t− 4)(t− 12) if t ≡ 12 mod 182t2−32t+102

9 = 29(t2 − 16t+ 51) if t ≡ 15 mod 18

0 if t 6≡ 0 mod 3

∑t≥0

M3(t) zt =8z15

(2z3 + 1

)(1− z3) (1− z6) (1− z9)


Fourier–Dedekind Sums joint withSinai Robins


−→ EhrP(z) := 1 +∑t≥1

LP(t) zt =h(z)

(1− zp)dimP+1

Natural ingredients of formulas for Ehrhart quasipolynomials:

sn (c1, . . . , cd; c) :=1

c

c−1∑k=1

e2πi n/c(1− e2πi c1/c

) (1− e2πi c2/c

)· · ·(1− e2πi cd/c

)Examples sn(c1; c) ∼

⌊nc

⌋




−→ EhrP(z) := 1 +∑t≥1

LP(t) zt =h(z)

(1− zp)dimP+1

Natural ingredients of formulas for Ehrhart quasipolynomials:

sn (c1, . . . , cd; c) :=1

c

c−1∑k=1


) (1− e2πi c2/c

)· · ·(1− e2πi cd/c

)Examples sn(c1; c) ∼

⌊nc

⌋

s0(c1, c2; c) ∼c−1∑k=1

⌊k c1c

⌋2

∼ Dedekind sum



sn (c1, . . . , cd; c) :=1

c

c−1∑k=1


) (1− e2πi c2/c

)· · ·(1− e2πi cd/c

)Fun Facts (Dedekind 1880s, Rademacher 1950s)

I sn(a, 1; b) = sn(a mod b, 1; b)

I sn(a, 1; b) + sn(b, 1; a) = something simple [reciprocity]



sn (c1, . . . , cd; c) :=1

c

c−1∑k=1


) (1− e2πi c2/c

)· · ·(1− e2πi cd/c


I sn(a, 1; b) = sn(a mod b, 1; b)


Corollary sn (c1, c2; c) can be computed in time log max(c1, c2, c).



sn (c1, . . . , cd; c) :=1

c

c−1∑k=1


) (1− e2πi c2/c

)· · ·(1− e2πi cd/c


I sn(a, 1; b) = sn(a mod b, 1; b)


Corollary sn (c1, c2; c) can be computed in time log max(c1, c2, c).

Corollary The Ehrhart quasipolynomial of any rational polygon P can beexpressed in terms of Fourier–Dedekind sums and computed in linear timein the input size of P.


Dedekind–Carlitz Polynomials joint withChristian HaaseAsia Matthews

Dedekind sum s0(a, 1; b) ∼b−1∑k=1

⌊ka

b

⌋2

∼b−1∑k=1

⌊ka

b

⌋(k − 1)

Dedekind–Carlitz Polynomial c (u, v; a, b) :=

b−1∑k=1

ubkab cvk−1



Dedekind sum s0(a, 1; b) ∼b−1∑k=1

⌊ka

b

⌋2

∼b−1∑k=1

⌊ka

b

⌋(k − 1)

Dedekind–Carlitz Polynomial c (u, v; a, b) :=

b−1∑k=1

ubkab cvk−1

Theorem (Carlitz 1975) If a and b are relatively prime,

(v − 1) c (u, v; a, b) + (u− 1) c (v, u; b, a) = ua−1vb−1 − 1

Applying u ∂u twice and v ∂v once gives Dedekind’s reciprocity law.



c (u, v; a, b) :=

b−1∑k=1

ubkab cvk−1

Consider the 2-dimensional cone

K := {λ1(0, 1) + λ2(a, b) : λ1, λ2 ≥ 0}

Fun Fact σK(u, v) :=∑

(m,n)∈K∩Z2

umvn =1 + uv c (v, u; b, a)

(1− v) (1− uavb)



c (u, v; a, b) :=

b−1∑k=1

ubkab cvk−1

Consider the 2-dimensional cone

K := {λ1(0, 1) + λ2(a, b) : λ1, λ2 ≥ 0}

Fun Fact σK(u, v) :=∑

(m,n)∈K∩Z2

umvn =1 + uv c (v, u; b, a)

(1− v) (1− uavb)

I Carlitz reciprocity = two cones adding up to R2≥0

I Complexity anyone?


A (Small) Bouquet of Open Problems

I Classify Ehrhart polynomials (or, alternatively, Ehrhart h-vectors)

I Find more inside-out polytopes

I Attack existence problems via (discrete-geometric) polynomials

I Study periods of Ehrhart quasipolynomials

I Study complexity of Fourier–Dedekind sums

I Compute vol(B11)


Birkhoff–von Neumann Revisited joint withDennis PixtonJesus De LoeraMike DevelinJulian PfeifleRichard Stanley

For more about roots of

(Ehrhart) polynomials,

see Braun (2008) and

Pfeifle (2010).


Documents

Discrete Volume Computations for Polyhedramath.sfsu.edu/beck/papers/nantes.pdf · Discrete Volume Computations for Polyhedra x y z 5 20 2 Matthias Beck San Francisco State University