Discrete Volume Computations. for Polyhedra

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1 Digital Discrete Volume Computations y for Polyhedra 20 Matthias Beck San Francisco State University math.sfsu.edu/beck DGCI x z 2

2 Science is what we understand well enough to explain to a computer, art is all the rest. Donald Knuth Discrete Volume Computations for Polyhedra Matthias Beck 2

3 Themes (0, 1, 3, 2) (1, 0, 3, 2) (1, 0, 2, 3) (0, 3, 2, 1) (0, 3, 1, 2) (3, 0, 2, 1) (3, 0, 1, 2) (1, 3, 0, 2) (0, 1, 2, 3) (1, 2, 0, 3) (3, 1, 0, 2) (1, 3, 2, 0) (0, 2, 1, 3) (1, 2, 3, 0) (3, 1, 2, 0) (2, 1, 0, 3) (0, 2, 3, 1) (2, 0, 1, 3) (2, 1, 3, 0) Discrete-geometric polynomials Computation (complexity) (2, 0, 3, 1) (3, 2, 1, 0) (3, 2, 0, 1) (2, 3, 1, 0) (2, 3, 0, 1) Generating functions Combinatorial structures Discrete Fourier analysis Discrete Volume Computations for Polyhedra Matthias Beck 3

4 A Sample Problem: Birkhoff von Neumann Polytope B n = x 11 x 1n.. x n1... x nn R n2 0 : j x jk = 1 for all 1 k n k x jk = 1 for all 1 j n Discrete Volume Computations for Polyhedra Matthias Beck 4

5 Discrete Volumes Rational polyhedron P R d solution set of a system of linear equalities & inequalities with integer coefficients Goal: understand P Z d... (list) z m1 1 zm 2 2 z m d d m P Z d (count) P Z d Discrete Volume Computations for Polyhedra Matthias Beck 5

6 Discrete Volumes Rational polyhedron P R d solution set of a system of linear equalities & inequalities with integer coefficients Goal: understand P Z d... (list) z m1 1 zm 2 2 z m d d m P Z d (count) P Z d 1 (volume) vol(p) = lim t t d P 1 t Zd Discrete Volume Computations for Polyhedra Matthias Beck 5

7 Discrete Volumes Rational polyhedron P R d solution set of a system of linear equalities & inequalities with integer coefficients Goal: understand P Z d... (list) z m1 1 zm 2 2 z m d d m P Z d (count) P Z d 1 (volume) vol(p) = lim t t d P 1 t Zd Ehrhart function L P (t) := P 1 t Zd = tp Z d for t Z >0 Discrete Volume Computations for Polyhedra Matthias Beck 5

8 Some Motivation Linear systems are everywhere, and so polyhedra are everywhere. Discrete Volume Computations for Polyhedra Matthias Beck 6

9 Some Motivation Linear systems are everywhere, and so polyhedra are everywhere. In applications, the volume of the polytope represented by a linear system measures some fundamental data of this system ( average ). Discrete Volume Computations for Polyhedra Matthias Beck 6

10 Some Motivation Linear systems are everywhere, and so polyhedra are everywhere. In applications, the volume of the polytope represented by a linear system measures some fundamental data of this system ( average ). Polytopes are basic geometric objects, yet even for these basic objects volume computation is hard and there remain many open problems. Discrete Volume Computations for Polyhedra Matthias Beck 6

11 Some Motivation Linear systems are everywhere, and so polyhedra are everywhere. In applications, the volume of the polytope represented by a linear system measures some fundamental data of this system ( average ). Polytopes are basic geometric objects, yet even for these basic objects volume computation is hard and there remain many open problems. Many discrete problems in various mathematical areas are linear problems, thus they ask for the discrete volume of a polytope in disguise. Discrete Volume Computations for Polyhedra Matthias Beck 6

12 A Warm-Up Ehrhart Function Lattice polytope P R d convex hull of finitely points in Z d For t Z >0 let L P (t) := # ( tp Z d) Example: = conv {(0, 0), (1, 0), (0, 1)} = { (x, y) R 2 0 : x + y 1} Discrete Volume Computations for Polyhedra Matthias Beck 7

13 A Warm-Up Ehrhart Function Lattice polytope P R d convex hull of finitely points in Z d For t Z >0 let L P (t) := # ( tp Z d) Example: = conv {(0, 0), (1, 0), (0, 1)} = { (x, y) R 2 0 : x + y 1} L (t) = ( ) t = 1 2 (t + 1)(t + 2) Discrete Volume Computations for Polyhedra Matthias Beck 7

14 A Warm-Up Ehrhart Function Lattice polytope P R d convex hull of finitely points in Z d For t Z >0 let L P (t) := # ( tp Z d) Example: = conv {(0, 0), (1, 0), (0, 1)} = { (x, y) R 2 0 : x + y 1} L (t) = ( ) t = 1 2 (t + 1)(t + 2), a polynomial in t with leading coefficient vol ( ) = 1 2 Discrete Volume Computations for Polyhedra Matthias Beck 7

15 A Warm-Up Ehrhart Function Lattice polytope P R d convex hull of finitely points in Z d For t Z >0 let L P (t) := # ( tp Z d) Example: = conv {(0, 0), (1, 0), (0, 1)} = { (x, y) R 2 0 : x + y 1} L (t) = ( ) t = 1 2 (t + 1)(t + 2) comes with the friendly generating function t 0 ( ) t + 2 z t = 2 1 (1 z) 3 Discrete Volume Computations for Polyhedra Matthias Beck 7

16 Ehrhart Polynomials Theorem (Ehrhart 1962) For any lattice polytope P, L P (t) is a polynomial in t of degree dim P with leading coefficient vol P and constant term 1. Equivalently, Ehr P (z) := 1 + t 1 L P (t) z t is rational: Ehr P (z) = h(z) (1 z) dim P+1 where the Ehrhart h-vector h(z) satisfies h(0) = 1 and h(1) = (dim P)! vol(p). Discrete Volume Computations for Polyhedra Matthias Beck 8

17 Ehrhart Polynomials Theorem (Ehrhart 1962) For any lattice polytope P, L P (t) is a polynomial in t of degree dim P with leading coefficient vol P and constant term 1. Equivalently, Ehr P (z) := 1 + t 1 L P (t) z t is rational: Ehr P (z) = h(z) (1 z) dim P+1 where the Ehrhart h-vector h(z) satisfies h(0) = 1 and h(1) = (dim P)! vol(p). Seeming dichotomy: vol(p) = lim via a finite amount of data. t 1 t dim P L P(t) can be computed discretely Discrete Volume Computations for Polyhedra Matthias Beck 8

18 Ehrhart Polynomials Theorem (Ehrhart 1962) For any lattice polytope P, L P (t) is a polynomial in t of degree d := dim P with leading coefficient vol P and constant term 1. Ehr P (z) := 1 + t 1 L P (t) z t = h(z) (1 z) d+1 Equivalent descriptions of an Ehrhart polynomial: L P (t) = c d t d + c d 1 t d c 0 via roots of L P (t) ( Ehr P (z) L P (t) = h t+d ) ( 0 d + t+d 1 ) ( h1 d + + t hd d) Discrete Volume Computations for Polyhedra Matthias Beck 8

19 Ehrhart Polynomials Theorem (Ehrhart 1962) For any lattice polytope P, L P (t) is a polynomial in t of degree d := dim P with leading coefficient vol P and constant term 1. Ehr P (z) := 1 + t 1 L P (t) z t = h(z) (1 z) d+1 ( L P (t) = h t+d ) ( 0 d + t+d 1 ) ( h1 d + + t hd d) Theorem (Macdonald 1971) ( 1) d L P ( t) enumerates the interior lattice points in tp. Equivalently, L P (t) = h d ( t+d 1 d ) + hd 1 ( t+d 2 d ) + + h0 ( t 1 d ) Discrete Volume Computations for Polyhedra Matthias Beck 9

20 Ehrhart Polynomials Theorem (Ehrhart 1962) For any lattice polytope P, L P (t) is a polynomial in t of degree d := dim P with leading coefficient vol P and constant term 1. Ehr P (z) := 1 + t 1 L P (t) z t = h(z) (1 z) d+1 L P (t) = h d ( t+d 1 d ) + hd 1 ( t+d 2 d ) + + h0 ( t 1 d ) Theorem (Stanley 1980) h 0, h 1,..., h d are nonnegative integers. Discrete Volume Computations for Polyhedra Matthias Beck 9

Ehrhart Polynomials Theorem (Ehrhart 1962) For any lattice polytope P, L P (t) is a polynomial in t of degree d := dim P with leading coefficient vol P and constant term 1.

21 Ehrhart Polynomials Theorem (Ehrhart 1962) For any lattice polytope P, L P (t) is a polynomial in t of degree d := dim P with leading coefficient vol P and constant term 1. Ehr P (z) := 1 + t 1 L P (t) z t = h(z) (1 z) d+1 L P (t) = h d ( t+d 1 d ) + hd 1 ( t+d 2 d ) + + h0 ( t 1 d ) Theorem (Stanley 1980) h 0, h 1,..., h d are nonnegative integers. Corollary If h d+1 k > 0 then kp contains an integer point. Discrete Volume Computations for Polyhedra Matthias Beck 9

22 Interlude: Graph Coloring a la Ehrhart ( ) k χ K2 (k) = k + 1 K 2 k + 1 x 1 = x 2 (Blass Sagan) Discrete Volume Computations for Polyhedra Matthias Beck 10

23 Interlude: Graph Coloring a la Ehrhart ( ) k χ K2 (k) = k + 1 K 2 k + 1 Similarly, for any given graph G on n nodes, we can write ( ) k + n χ G (k) = a 0 n x 1 = x 2 ( ) k + n 1 + a 1 n + + a n ( k n (Blass Sagan) ) for some (meaningful) nonnegative integers a 0,..., a n Discrete Volume Computations for Polyhedra Matthias Beck 10

24 Computational Complexity of Integer-Point Transforms Rational polyhedron P R d solution set of a system of linear equalities & inequalities with integer coefficients σ P (z) := z m1 1 zm 2 2 z m d d m P Z d is a rational function in z 1, z 2,..., z d Lenstra (1983) polynomial-time algorithm to decide whether σ P (z) = 0 Barvinok (1994) polynomial-time algorithm to compute σ P (z) Discrete Volume Computations for Polyhedra Matthias Beck 11

25 Computational Complexity of Integer-Point Transforms Rational polyhedron P R d solution set of a system of linear equalities & inequalities with integer coefficients σ P (z) := z m1 1 zm 2 2 z m d d m P Z d is a rational function in z 1, z 2,..., z d Lenstra (1983) polynomial-time algorithm to decide whether σ P (z) = 0 Barvinok (1994) polynomial-time algorithm to compute σ P (z) Example P = [0, 1000] σ [0,1000] (z) = 1 + z + + z 1000 = 1 z z Discrete Volume Computations for Polyhedra Matthias Beck 11

26 Computational Complexity of Integer-Point Transforms Rational polyhedron P R d solution set of a system of linear equalities & inequalities with integer coefficients σ P (z) := z m1 1 zm 2 2 z m d d m P Z d is a rational function in z 1, z 2,..., z d Lenstra (1983) polynomial-time algorithm to decide whether σ P (z) = 0 Barvinok (1994) polynomial-time algorithm to compute σ P (z) Given a polytope P we can compute Ehr P (z) = σ cone(p) (1, 1,..., 1, z) where cone(p) := R 0 (P {1}) Discrete Volume Computations for Polyhedra Matthias Beck 11

27 Computational Complexity of Integer-Point Transforms Rational polyhedron P R d solution set of a system of linear equalities & inequalities with integer coefficients σ P (z) := z m1 1 zm 2 2 z m d d m P Z d is a rational function in z 1, z 2,..., z d Lenstra (1983) polynomial-time algorithm to decide whether σ P (z) = 0 Barvinok (1994) polynomial-time algorithm to compute σ P (z) Implementations: De Loera, Köppe et al latte Verdoolaege freshmeat.net/projects/barvinok Discrete Volume Computations for Polyhedra Matthias Beck 11

28 Ehrhart Quasipolynomials Rational polytope P R d convex hull of finitely points in Q d Theorem (Ehrhart 1962) L P (t) is a quasipolynomial in t: L P (t) = c d (t) t d + c d 1 (t) t d c 0 (t) where c 0 (t),..., c d (t) are periodic functions. Example L [0, 1 2 ] (t) = 1 2 t ( 1)t 4 Discrete Volume Computations for Polyhedra Matthias Beck 12

29 Ehrhart Quasipolynomials Rational polytope P R d convex hull of finitely points in Q d Theorem (Ehrhart 1962) L P (t) is a quasipolynomial in t: L P (t) = c d (t) t d + c d 1 (t) t d c 0 (t) where c 0 (t),..., c d (t) are periodic functions. Equivalently, Ehr P (z) := 1 + t 1 L P (t) z t = h(z) (1 z p ) dim P+1 for some (minimal) p Z >0 (the period of L P (t)). Example Ehr [0, 1 2 ] (z) = t 0 ( 1 2 t ) ( 1)t z t = z (1 z 2 ) 2 Discrete Volume Computations for Polyhedra Matthias Beck 12

30 Interlude: Magic Squares M n (t) number of n n squares with distinct entries and row, column, and diagonal sums t joint with Thomas Zaslavsky Andrew Van Herick Similar to graph coloring, this is a polyhedral problem with forbidden hyperplanes: inside-out polytopes Discrete Volume Computations for Polyhedra Matthias Beck 13

31 Interlude: Magic Squares joint with Thomas Zaslavsky Andrew Van Herick M n (t) number of n n squares with distinct entries and row, column, and diagonal sums t Example M 3 (t) = 2t 2 32t = 2 9 (t2 16t + 72) if t 0 mod 18 (t 3)(t 13) if t 3 mod 18 2t 2 32t+78 9 = 2 9 2t 2 32t = 2 9 2t 2 32t = 2 9 2t 2 32t+96 9 = 2 9 (t 6)(t 10) if t 6 mod 18 (t 7)(t 9) if t 9 mod 18 (t 4)(t 12) if t 12 mod 18 2t 2 32t = 2 9 (t2 16t + 51) if t 15 mod 18 0 if t 0 mod 3 M 3 (t) z t = t 0 8z 15 ( 2z ) (1 z 3 ) (1 z 6 ) (1 z 9 ) Discrete Volume Computations for Polyhedra Matthias Beck 13

32 Fourier Dedekind Sums joint with Sinai Robins Rational polytope P R d convex hull of finitely points in Q d Ehr P (z) := 1 + t 1 L P (t) z t = h(z) (1 z p ) dim P+1 Natural ingredients of formulas for Ehrhart quasipolynomials: s n (c 1,..., c d ; c) := 1 c Examples s n (c 1 ; c) c 1 k=1 n c 2πi n/c e ( ) ( 1 e 2πi c 1 /c 1 e ) 2πi c 2/c (1 e ) 2πi c d/c Discrete Volume Computations for Polyhedra Matthias Beck 14

33 Fourier Dedekind Sums joint with Sinai Robins Rational polytope P R d convex hull of finitely points in Q d Ehr P (z) := 1 + t 1 L P (t) z t = h(z) (1 z p ) dim P+1 Natural ingredients of formulas for Ehrhart quasipolynomials: s n (c 1,..., c d ; c) := 1 c c 1 k=1 2πi n/c e ( ) ( 1 e 2πi c 1 /c 1 e ) 2πi c 2/c (1 e ) 2πi c d/c Examples s n (c 1 ; c) n c s 0 (c 1, c 2 ; c) c 1 k=1 2 k c1 Dedekind sum c Discrete Volume Computations for Polyhedra Matthias Beck 14

34 Fourier Dedekind Sums joint with Sinai Robins s n (c 1,..., c d ; c) := 1 c c 1 k=1 Fun Facts (Dedekind 1880s, Rademacher 1950s) s n (a, 1; b) = s n (a mod b, 1; b) 2πi n/c e ( ) ( 1 e 2πi c 1 /c 1 e ) 2πi c 2/c (1 e ) 2πi c d/c s n (a, 1; b) + s n (b, 1; a) = something simple [reciprocity] Discrete Volume Computations for Polyhedra Matthias Beck 15

35 Fourier Dedekind Sums joint with Sinai Robins s n (c 1,..., c d ; c) := 1 c c 1 k=1 Fun Facts (Dedekind 1880s, Rademacher 1950s) s n (a, 1; b) = s n (a mod b, 1; b) 2πi n/c e ( ) ( 1 e 2πi c 1 /c 1 e ) 2πi c 2/c (1 e ) 2πi c d/c s n (a, 1; b) + s n (b, 1; a) = something simple [reciprocity] Corollary s n (c 1, c 2 ; c) can be computed in time log max(c 1, c 2, c). Discrete Volume Computations for Polyhedra Matthias Beck 15

36 Fourier Dedekind Sums joint with Sinai Robins s n (c 1,..., c d ; c) := 1 c c 1 k=1 Fun Facts (Dedekind 1880s, Rademacher 1950s) s n (a, 1; b) = s n (a mod b, 1; b) 2πi n/c e ( ) ( 1 e 2πi c 1 /c 1 e ) 2πi c 2/c (1 e ) 2πi c d/c s n (a, 1; b) + s n (b, 1; a) = something simple [reciprocity] Corollary s n (c 1, c 2 ; c) can be computed in time log max(c 1, c 2, c). Corollary The Ehrhart quasipolynomial of any rational polygon P can be expressed in terms of Fourier Dedekind sums and computed in linear time in the input size of P. Discrete Volume Computations for Polyhedra Matthias Beck 15

37 Dedekind Carlitz Polynomials joint with Christian Haase Asia Matthews Dedekind sum s 0 (a, 1; b) b 1 k=1 2 ka b b 1 k=1 ka (k 1) b Dedekind Carlitz Polynomial c (u, v; a, b) := b 1 k=1 u ka b v k 1 Discrete Volume Computations for Polyhedra Matthias Beck 16

38 Dedekind Carlitz Polynomials joint with Christian Haase Asia Matthews Dedekind sum s 0 (a, 1; b) b 1 k=1 2 ka b b 1 k=1 ka (k 1) b Dedekind Carlitz Polynomial c (u, v; a, b) := b 1 k=1 u ka b v k 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) = u a 1 v b 1 1 Applying u u twice and v v once gives Dedekind s reciprocity law. Discrete Volume Computations for Polyhedra Matthias Beck 16

39 Dedekind Carlitz Polynomials joint with Christian Haase Asia Matthews c (u, v; a, b) := b 1 k=1 u ka b v k 1 Consider the 2-dimensional cone K := {λ 1 (0, 1) + λ 2 (a, b) : λ 1, λ 2 0} Fun Fact σ K (u, v) := (m,n) K Z 2 u m v n = 1 + uv c (v, u; b, a) (1 v) (1 u a v b ) Discrete Volume Computations for Polyhedra Matthias Beck 17

40 Dedekind Carlitz Polynomials joint with Christian Haase Asia Matthews c (u, v; a, b) := b 1 k=1 u ka b v k 1 Consider the 2-dimensional cone K := {λ 1 (0, 1) + λ 2 (a, b) : λ 1, λ 2 0} Fun Fact σ K (u, v) := (m,n) K Z 2 u m v n = 1 + uv c (v, u; b, a) (1 v) (1 u a v b ) Carlitz reciprocity = two cones adding up to R 2 0 Complexity anyone? Discrete Volume Computations for Polyhedra Matthias Beck 17

41 A (Small) Bouquet of Open Problems Classify Ehrhart polynomials (or, alternatively, Ehrhart h-vectors) Find more inside-out polytopes Attack existence problems via (discrete-geometric) polynomials Study periods of Ehrhart quasipolynomials Study complexity of Fourier Dedekind sums Compute vol(b 11 ) Discrete Volume Computations for Polyhedra Matthias Beck 18

42 Birkhoff von Neumann Revisited joint with Dennis Pixton Jesus De Loera Mike Develin Julian Pfeifle Richard Stanley For more about roots of (Ehrhart) polynomials, see Braun (2008) and Pfeifle (2010). Discrete Volume Computations for Polyhedra Matthias Beck 19

Classification of Ehrhart quasi-polynomials of half-integral polygons

Classification of Ehrhart quasi-polynomials of half-integral polygons A thesis presented to the faculty of San Francisco State University In partial fulfilment of The Requirements for The Degree Master