McMaster University. Advanced Optimization Laboratory. Advanced Optimization Laboratory. Title: Title: Small Primitive strings Zonotopes

Transcription

1 McMaster University Avance Optimization Laboratory Avance Optimization Laboratory Title: Title: A computational framework for etermining square-maximal Small Primitive strings Zonotopes Authors: Authors: Antoine Deza, George Manoussakis, an Shmuel Onn Antoine Deza, Frantisek Franek, an Mei Jiang AvOL-Report No. 2011/5 AvOL-Report No. 2017/1 December 2011, Hamilton, Ontario, Canaa February 2017, Hamilton, Ontario, Canaa

2 Small Primitive Zonotopes Antoine Deza a, George Manoussakis b, an Shmuel Onn c a McMaster University, Hamilton, Ontario, Canaa b Université e Paris Su, Orsay, France c Technion - Israel Institute of Technology, Haifa, Israel Abstract We stuy a family of lattice polytopes, calle primitive zonotopes, which can be seen as a generalization of the permutaheron of type B. We escribe primitive zonotopes with small parameters, an iscuss connections to the largest iameter of lattice polytopes an to the computational complexity of multicriteria matroi optimization. Complexity results an open questions are also presente. Keywors: Lattice polytopes, primitive integer vectors, matroi optimization, iameter 1 Introuction We pursue the stuy of the primitive zonotopes initiate in [7]. After recalling their efinition an some of their properties, we highlight in Section 2 connections between primitive zonotopes an convex matroi optimization an the iameter of lattice polytopes. Small instances are presente in Section 3, an complexity results an open questions are iscusse in Section 4. Recent results ealing with the combinatorial, geometric, an algorithmic aspects of linear optimization inclue Santos counterexample [24] to the Hirsch conjecture, an Allamigeon, Benchimol, Gaubert, an Joswig s counterexample [2] to a continuous analogue of the polynomial Hirsch conjecture. The Hirsch boun was valiate by Borgwart, De Loera, an Finhol [4]. Kalai an Kleitman s upper boun [15] for the iameter of polytopes was strengthene by To [29], an then by Sukegawa [27]. Kleinschmit an Onn s upper boun [16] for the iameter of lattice polytopes was strengthene by Del Pia an Michini [6], an then by Deza an Pournin [8]. Multicriteria matroi optimization is a generalization of stanar linear matroi optimization introuce by Onn an Rothblum [23] where each basis is evaluate accoring to several, rather than one, criteria, an these values are trae-in by a convex function. 1

3 2 2 Primitive zonotopes 2.1 Zonotopes generate by short primitive vectors The convex hull of integer-value points is calle a lattice polytope an, if all the vertices are rawn from {0, 1,..., k}, is referee to as a lattice (, k-polytope. For simplicity, we only consier full imensional lattice (, k-polytopes. Given a finite set G of vectors, also calle the generators, the zonotope generate by G is the convex hull of all signe sums of the elements of G. We consier zonotopes generate by short integer vectors in orer to keep the gri embeing size relatively small. In aition, we restrict to integer vectors which are pairwise linearly inepenent in orer to maximize the iameter. Thus, for q = or a positive integer, an, p positive integers, we consier the primitive zonotope Z q (, p efine as the zonotope generate by the primitive integer vectors of q-norm at most p: Z q (, p = [ 1, 1]{v Z : v q p, gc(v = 1, v 0} ( = conv {λv v : v Z, v q p, gc(v = 1, v 0} : λ v = ±1 where gc(v is the largest integer iviing all entries of v, an the lexicographic orer on R, i.e. v 0 if the first nonzero coorinate of v is positive. Similarly, we consier the Minkowski sum H q (, p of the generators of Z q (, p: H q (, p = [0, 1]{v Z : v q p, gc(v = 1, v 0}. In other wors, H q (, p is, up to translation, the image of Z q (, p by a homothety of factor 1/2. We also consier the positive primitive zonotope Z + q (, p efine as the zonotope generate by the primitive integer vectors of q-norm at most p with nonnegative coorinates: Z + q (, p = [ 1, 1]{v Z + : v q p, gc(v = 1} where Z + = {0, 1,... }. Similarly, we consier the Minkowski sum of the generators of Z + q (, p: H + q (, p = [0, 1]{v Z + : v q p, gc(v = 1}. We illustrate the primitive zonotopes with a few examples: (i For finite q, Z q (, 1 is generate by the unit vectors an forms the { 1, 1} -cube. H q (, 1 is the {0, 1} -cube. (ii Z 1 (, 2 is the permutaheron of type B an thus, H 1 (, 2 is, up to translation, a lattice (, 2 1-polytope with 2! vertices an iameter 2. For example, Z 1 (2, 2 is generate by {(0, 1, (1, 0, (1, 1, (1, 1} an forms the octagon whose vertices are {( 3, 1, ( 3, 1, ( 1, 3, (1, 3, (3, 1, (3, 1, (1, 3, ( 1, 3}. H 1 (2, 2 is, up to translation, a lattice (2, 3-polygon, see Figure 1. Z 1 (3, 2 is congruent to

4 3 Figure 1: H 1 (2, 2 the truncate cuboctaheron, see Figure 2 for an illustration, which is also calle the great rhombicuboctaheron an is the Minkowski sum of an octaheron an a cuboctaheron, see for instance Eppstein [9]. H 1 (3, 2 is, up to translation, a lattice (3, 5-polytope with iameter 9 an 48 vertices. Figure 2: Z 1 (3, 2 is congruent to the truncate cuboctaheron (iii H 1 + (, 2 is the Minkowski sum of the permutaheron with the {0, 1} -cube. Thus, H 1 + (, 2 is a lattice (, -polytope with iameter ( (iv Z (3, 1 is congruent to the truncate small rhombicuboctaheron, see Figure 3 for an illustration, which is the Minkowski sum of a cube, a truncate octaheron, an a rhombic oecaheron, see for instance Eppstein [9]. H (3, 1 is, up to translation, a lattice (3, 9-polytope with iameter 13 an 96 vertices. (v Z + (2, 2 is generate by {(0, 1, (1, 0, (1, 1, (1, 2, (2, 1} an forms the ecagon whose vertices are {( 5, 5, ( 5, 3, ( 3, 5, ( 3, 1, ( 1, 3, (1, 3, (3, 1, (3, 5, (5, 3, (5, 5}. H + (2, 2 is a lattice (2, 5-polygon, see Figure 4.

5 4 Figure 3: Z (3, 1 is congruent to the truncate small rhombicuboctaheron Figure 4: H + (2, Combinatorial properties of the primitive zonotopes We recall properties concerning Z q (, p an Z q + (, p, an in particular their symmetry group, iameter, an vertices. Z 1 (, 2 is the permutaheron of type B as its generators form the root system of type B, see [14]. Thus, Z 1 (, 2 has 2! vertices an its symmetry group is B. The properties liste in this section are extensions to Z q (, p of known properties of Z 1 (, 2 whose proofs are given in Section 5.1. We refer to Fukua [11], Grünbaum [13], an Ziegler [30] for polytopes an, in particular, zonotopes. Property 2.1. (i Z q (, p is invariant uner the symmetries inuce by coorinate permutations an the reflections inuce by sign flips. (ii The sum σ q (, p of all the generators of Z q (, p is a vertex of both Z q (, p an H q (, p. The origin is a vertex of H q (, p, an σ q (, p is a vertex of Z q (, p. (iii The coorinates of the vertices of Z q (, p are o. Thus, the number of vertices of Z q (, p is a multiple of 2.

6 5 (iv H q (, p is, up to translation, a lattice (, k-polytope where k is the sum of the first coorinates of all generators of Z q (, p (v The iameter of Z q (, p, respectively Z + q (, p, is equal to the number of its generators. Property 2.2. (i Z + q (, p is centrally symmetric an invariant uner the symmetries inuce by coorinate permutations. (ii The sum σ + q (, p of all the generators of Z + q (, p is a vertex of both Z + q (, p an H + q (, p. The origin is a vertex of H + q (, p, an σ + q (, p is a vertex of Z + q (, p. A vertex v of Z q (, p is calle canonical if v 1 v > 0. Property 2.1 item (i implies that the vertices of Z q (, p are all the coorinate permutations an sign flips of its canonical vertices.. Property 2.3. (i A canonical vertex v of Z q (, p is the unique maximizer of {max c T x : x Z q (, p} for some vector c satisfying c 1 > c 2 > > c > 0. (ii Z 1 (, 2 has 2! vertices corresponing to all coorinate permutations an sign flips of the unique canonical vertex σ 1 (, 2 = (2 1, 2 3,..., 1. (iii Z q (, p has at least 2! vertices incluing all coorinate permutations an sign flips of the canonical vertex σ q (, p. (iv Z + (, 1 has at least 2 + 2! vertices incluing the 2! permutations of ±σ( where σ( is a vertex with pairwise istinct coorinates, an the 2 vertices ±σ + (, Primitive zonotopes as lattice polytopes with large iameter Let δ(, k be the maximum possible ege-iameter over all lattice (, k-polytopes. Fining lattice polygons with the largest iameter; that is, to etermine δ(2, k, was investigate inepenently in the early nineties by Thiele [28], Balog an Bárány [3], an Acketa an Žunić [1]. This question can be foun in Ziegler s book [30] as Exercise The answer is summarize in Proposition 2.4, with the role of primitive zonotopes highlighte. Proposition 2.4. δ(2, k is achieve, up to translation, by the Minkowski sum of a subset of the generators of H 1 (2, p for a proper p. In particular, for k = jφ(j for some p, δ(2, k is uniquely achieve, up to translation, by H 1 (2, p. 1 j p

7 6 In general imension, Naef [21] showe in 1989 that δ(, 1 =, Kleinschmit an Onn [16] generalize this result in 1992 showing that δ(, k k, before Del Pia an Michini [6] strengthene the upper boun to δ(, k k /2 for k 2, an showe that δ(, 2 = 3/2, an Deza an Pournin [8]. further strengthene the upper boun to k 2/3 (k 3 for k 3 an showe that δ(3, 4 = 8. Concerning the lower boun Deza, Manoussakis an Onn [7] showe that δ(, k (k + 1/2 for k < 2. These bouns are summarize in Proposition 2.5, an Conjecture 2.6 given in [7] is recalle. Proposition 2.5. (i δ(, k = (k + 1/2 for (, k = (, 1, (, 2, (2, 3, (3, 3, an (3, 4, (ii 2 δ(, 3 7/3 1 for 2 mo 3, an δ(, 3 7/3 otherwise, (iii δ(, k (k + 1/2 for k < 2, (iv k 2/3 (k 2 for k 4 Conjecture 2.6. δ(, k (k + 1/2, an δ(, k is achieve, up to translation, by a Minkowski sum of lattice vectors. Note that Conjecture 2.6 hols for all known values of δ(, k given in Table 1, an hypothesizes, in particular, that δ(, 3 = 2. k δ(, k /2 Table 1: Largest iameter δ(, k over all lattice (, k-polytopes Soprunov an Soprunova [26] consiere the Minkowski length of a lattice polytope P ; that is, the largest number of lattice segments whose Minkowski sum is containe in P. For example, the Minkowski length of the {0, k} -cube is k. We consier a variant of the Minkowski length an the special case when P is the {0, k} -cube. Let L(, k enote the largest number of pairwise linearly inepenent lattice segments whose Minkowski sum is containe in the {0, k} -cube. One can check that the generators of H 1 (, 2 form the largest, an unique, set of primitive lattice vectors which Minkowski sum fits within the {0, k} -cube for k = 2 1; that is, for k being the sum of the first coorinates of the 2 generators of H 1 (, 2. Thus, L(, 2 1 = δ(h 1 (, 2 = 2. Similarly, L(2, k = δ(2, k for all k, an L(, k = (k + 1/2 for k 2 1.

8 7 2.4 Primitive zonotope an convex matroi optimization We consier the convex multicriteria matroi optimization framework of Melame, Onn an Rothblum, see [19, 22, 23]. Call S {0, 1} n a matroi if it is the set of the inicators of bases of a matroi over {1,..., n}. For instance, S can be the set of inicators of spanning trees in a connecte graph with n eges. For a n matrix W, let W S = {W x : x S}, an let conv(w S = W conv(s be the projection to R of conv(s by W. Given a convex function f : R R, convex matroi optimization eals with maximizing the composite function f(w x over S; that is, max {f(w x : x S}, an is concerne with conv(w S; that is, the projection of the set of the feasible points. The maximization problem can be interprete as a problem of multicriteria optimization, where each row of W gives a linear criterion W i x an f compromises these criteria. Thus, W is calle the criteria matrix or weight matrix. The projection polytope conv(w S an its vertices play a key role in solving the maximization problem as, for any convex function f, there is an optimal solution x whose projection y = W x is a vertex of conv(w S. In particular, the enumeration of all vertices of conv(w S enables to compute the optimal objective value by picking a vertex attaining the optimal value f(y = f(w x. Thus, it suffices that f is presente by a comparison oracle that, querie on vectors y, z R, asserts whether or not f(y < f(z. Coarse criteria matrices; that is, W whose entries are small or in {0, 1,..., p}, are of particular interest. In multicriteria combinatorial optimization, this case correspons to the weight W i,j attribute to element j of the groun set {1,..., n} uner criterion i being small or in {0, 1,..., p} for all i, j. In the remainer, we only consier {0, 1,..., p}-value W. Let m(, p enote the number of vertices of H (, p. Theorem 2.7, given in [7], settles the computational complexity of the multicriteria optimization problem by showing that the maximum number of vertices of the projection polytope conv(w S of any matroi S on n elements an any -criteria p-boune utility matrix; that is, W {0, 1,..., p} n, is equal to m(, p, an hence is in particular inepenent of n, S, an W. Theorem 2.7. Let, p be any positive integers. Then, for any positive integer n, any matroi S {0, 1} n, an any -criteria p-boune utility matrix W, the primitive zonotope H (, p refines conv(w S. Moreover, H (, p is a translation of conv(w S for some matroi S an -criteria p-boune utility matrix W. Thus, the maximum number of vertices of conv(w S for any n, any matroi S {0, 1} n, an any -criteria p-boune utility matrix W, equals the number m(, p of vertices of H (, p, an hence is inepenent of n, S, an W. Also, for any fixe an convex f : R R, the multicriteria matroi optimization problem can be solve using a number of arithmetic operations an queries to the oracles of S an f which is polynomial in n an p using m(, p greeily solvable linear matroi optimization counterparts. Theorem 2.8. The number m(, 1 of vertices of H (, 1 satisfies 2! m(, 1 2 ( ( (3 3/2 (3 1 3/2 2. i 1 0 i 1

9 8 Proof. The first inequality restates item (iii of Property 2.3 where (q,, p is set to (,, 1. The secon inequality is obtaine by exploiting the structure of the generators of H (, 1. One can check that H (, 1 has (3 1/2 generators an that removing the first zero of the generators of H (, 1 starting with zero yiels exactly the (3 1 1/2 generators of H ( 1, 1. We recall that the number of vertices f 0 (Z of a -imensional zonotope Z generate by m generators is boune by f(, m = 2 0 i 1 ( m 1 i. By uality, the number f 0 (Z of vertices of a zonotope Z is equal to the number f 1 (A of cells of the associate hyperplane arrangement A where each generator m j of Z correspons to an hyperplane h j of A. The inequality f 0 (Z f(, m is base on the inequality f 1 (A f 1 (A \ h j + f 1 (A h j for any hyperplane h j of A where A \ h j enotes the arrangement obtaine by removing h j from A, an A h j enotes the arrangement obtaine by intersecting A with h j. This last inequality an the uality between zonotopes an hyperplane arrangements are etaile, for example, in [11]. Recursively applying this inequality to the arrangement A (, 1 associate to H (, 1 till the remaining (3 1 1/2 hyperplanes form a ( 1-imensional arrangement equivalent to A ( 1, 1 yiels: f 1 (A (, 1 f(, (3 1/2 ( f(, (3 1 1/2 f( 1, (3 1 1/2 which completes the proof since f 1 (A (, 1 = f 0 (H (, 1 an f(, m f( 1, m = 2 ( m 1. In other wors, the inequality is base on the inuctive buil-up of H (, 1 starting with the (3 1 3/2 generators with zero as first coorinate, an noticing that these (3 1 3/2 generators belong to a lower imensional space. 3 Small primitive zonotopes H q (, p an H + q (, p In this section we provie the number of vertices, the iameter; that is, the number of generators, an the gri embeing size for H q (, p an H q + (, p for small an p, an q = 1, 2, an. We recall that, up to translation, Z q (, p, respectively Z q + (, p, is the image of H q (, p, respectively H q + (, p, by a homothety of factor 2. Thus Z q (, p an H q (, p, respectively Z q + (, p an H q + (, p, have the same number of vertices an the same iameter, while the gri embeing size of the Z q (, p, respectively Z q + (, p, is twice the one of H q (, p, respectively H q + (, p. Since both H q (, 1 an H q + (, 1 are equal to the {0, 1} -cube for finite q, both are omitte from the tables provie in this section. The vertex enumeration was performe using stanar algorithms escribe, for instance, in [11]. The Euler totient function counting positive integers less than or equal to j an relatively prime with j is enote by φ(j. Note that φ(1 is set to 1. Enumerative questions concerning H q (, p an H + q (, p have been stuie in various settings. We list a few instances, an the associate OEI sequences, see [25] for etails an references therein. (i f 0 (H + (, 1 correspons to the OEI sequence A giving the number of generalize retare functions in quantum fiel theory. The value of f 0 (H + (, 1 was etermine till = 8.

10 9 (ii f 0 (H (, 1, which is the number of regions of hyperplane arrangements with { 1, 0.1}-value normals in imension, correspons to the OEI sequence A giving f 0 (H (, 1/(2!. The value of f 0 (H (, 1 was etermine till = 7. (iii δ(h + (, p correspons to the OEI sequence A with further cross-reference sequences for 7 an p 8. (iv δ(h + 1 (3, p, respectively δ(h + 2 (2, p, δ(h (, 2, δ(h (2, p/4, δ(h 2 (2, p/2, δ(h + 1 (, 3, an δ(h + 2 (, 2, correspons to the OEI sequence A048134, respectively A049715, A005059, A002088, A175341, A008778, an A (v the gri embeing size of H 2 (, 2, respectively H (, 2 an H + 1 (, 3, correspons to the OEI sequence A161712, respectively A an A Small primitive zonotopes H q (, p In Tables 2, 3, an 4, the number of vertices f 0 (H q (, p is ivie by 2! an followe by its iameter δ(h q (, p an gri embeing size. For instance, the entry 26(49, 53 for (q,, p = (1, 3, 4 in Table 2 inicates that H 1 (3, 4 has ! = 1248 vertices, iameter 49, an is, up to translation, a lattice (3, 53-polytope. The rather straightforwar proofs are given in Section Small primitive zonotopes H 1 (, p Property 3.1. (i H 1 (, 1 is the {0, 1} -cube, (ii H 1 (, 2 is, up to translation, a lattice (, k-polytope with k = 2 1, an iameter 2, an 2! vertices, (iii H 1 (, 3 is, up to translation, a lattice (, k-polytope with k = , an iameter ( + 2(2 1/3, (iv H 1 (, 4 is, up to translation, a lattice (, k-polytope with k = ( ( ( ( , an iameter ( /3, (v H 1 (2, p is, up to translation, a lattice (2, k-polygon with k = jφ(j, an 1 j p iameter 2 φ(j. 1 j p

11 10 p H 1 (, p (4,3 2 (8,9 3 (12,17 5 (20,37 6 (24, (9,5 7 (25,21 26 (49, (97, (145, (16,7 40 (56, (136, (312,337? (560, (25,9 339 (105,57? (305,217? (801,713? (1 681,1 769 Table 2: Small primitive zonotopes H 1 (, p Small primitive zonotopes H 2 (, p Property 3.2. (i H 2 (, 1 is the {0, 1} -cube, (ii H 2 (, 2 is, up to translation, a lattice (, k-polytope with k = iameter 2 j( j+1. 0 j 3 0 j 3 2 j( 1 j, an p H 2 (, p (4,3 2 (8,9 4 (16,27 6 (24, (13,9 26 (49, (109, (205, (40, (192,193? (592,795? (1 424, (105,65? (641,577 Table 3: Small primitive zonotopes H 2 (, p Small primitive zonotopes H (, p Property 3.3. (i H (, 1 is, up to translation, a lattice (, k-polytope with k = 3 1, an iameter (3 1/2, (ii H (, 2 is, up to translation, a lattice (, k-polytope with k = , an iameter (5 3 /2, (iii H (2, p is, up to translation, a lattice (2, k-polygon with iameter 4 φ(j. 1 j p

12 11 p H (, p (4,3 2 (8,9 4 (16,27 6 (24, (13,9 26 (49, (145, (289, (40, (272,321? (1 120,1 923? (2 928, (121, (364, (1 093,729 Table 4: Small primitive zonotopes H (, p 3.2 Small positive primitive zonotopes H + q (, p In Tables 5, 6, an 7, the number of vertices f 0 (H + q (, p is followe by its iameter δ(h + q (, p an gri embeing size. For instance, the entry 1082 (15, 5 for (q,, p = (1, 5, 2 in Table 5 inicates that H + 1 (5, 1 has vertices, iameter 15, an is a lattice (5, 5-polytope Small positive primitive zonotopes H + 1 (, p Property 3.4. (i H + 1 (, 1 is the {0, 1} -cube, (ii H + 1 (, 2 is a lattice (, k-polytope with k =, an iameter ( +1 2 (iii H 1 + (, 3 is a lattice (, k-polytope with k = ( /2 an iameter ( /6. (iv H 1 + (2, p is a lattice (2, k-polygon with k = 1 + jφ(j/2, an iameter j p φ(j. 1 j p, p H 1 + (, p (3,2 10 (5,5 14 (7,9 22 (11,19 26 (13, (6,3 110 (13, (22, (40, (55, (10, (26, (51, (103, (161, (15, (45, (100,67? (221,188? (386, (21,6? (71,31?(176,106 Table 5: Small positive primitive zonotopes H + 1 (, p

13 Small positive primitive zonotopes H + 2 (, p Property 3.5. (i H + 2 (, 1 is the {0, 1} -cube, (ii H + 2 (, 2 is a (, k polytope with k = ( 1 + ( 3, an iameter ( ( p H 2 + (, p (3,2 10 (5,5 18 (9,14 26 (13, (7,4 212 (19, (40, (70, (15, (55, (141, (299, (30,15? (136,108? (441,487 Table 6: Small positive primitive zonotopes H + 2 (, p Small positive primitive zonotopes H (, + p Property 3.6. (i H (, + 1 is, a lattice (, k-polytope with k = 2 1, an iameter 2 1, (ii H (, + 2 is a lattice (, k-polytope with k = 3 2, an iameter 3 2, (iii H (2, + p is a lattice (2, k-polygon with iameter φ(j. 1 j p p H (, + p (3,2 10 (5,5 18 (9,14 26 (13, (7,4 212 (19, (49, (91, (15, (65, (225,344? (529, (31,16? (211,211? (961,1456? (2 851, (63, (127, (255,128 Table 7: Small positive primitive zonotopes H + (, p

14 13 4 Complexity issues We iscuss a few complexity issues relate to primitive zonotopes. While we mainly focus on Z q (, p, the iscussion an results, such as Propositions 4.1 an 4.2, can be aapte to Z q + (, p. As H q (, p, respectively H q + (, p, is the translation of the image by a homothety of Z q (, p, respectively Z q + (, p, the complexity results are the same. 4.1 Complexity properties Proposition 4.1. For fixe positive integers p an q, linear optimization over Z q (, p is polynomial-time solvable, even in variable imension. Proof. Since the q-norm of a generator of Z q (, p is boune by p, it has at most p q nonzero entries which is attaine for the vector of all ones an = p q. Thus, the number of generators of Z q (, p is boune by ( p (2p p q = O ( pq. Hence, one can q explicitly write all the generators of Z q (, p in polynomial time. Consequently, one can compute in polynomial time the following signe sum of generators of Z q (, p for any given rational c R : v = sign(c T vv where G q (, p enotes the set of generators of v G q(,p Z q (, p. Note that sign(0 is set to 0. Then, one can show that v is a maximizer of {max c T x : x Z q (, p}. The algorithmic theory evelope by Grötschel, Lovász, an Schrijver [12] shows that polynomial-time solvability for linear optimization over a polytope implies polynomialtime solvability for other questions. In particular, Proposition 4.1 implies Proposition 4.2. Proposition 4.2. For fixe positive integers p an q, the following problems are polynomialtime solvable. (i Extremality: Given v Z, ecie if v is a vertex of Z q (, p, (ii Ajacency: Given v 1, v 2 Z, ecie if [v 1, v 2 ] is an ege of Z q (, p; (iii Separation: Given rational y R, either assert y Z q (, p, or fin h Z separating y from Z q (, p; that is, satisfying h T y > h T x for all x Z q (, p. 4.2 Open problems A natural open problem is to fin irect algorithms to solve, over both Z q (, p an Z + q (, p, the extremality, ajacency, an separation questions given in Proposition 4.2. Note that the case q =, even for p = 1, seems to be significantly harer as the number of nonzero entries in a generator of Z (, p can not boune by a constant inepenent of. Thus, the number of generators of Z (, p is exponential in. Hence, the complexity of linear optimization, extremality, ajacency, an separation over both Z (, p an Z + (, p, is open. In particular, it is not clear if eciing if a given point is a vertex

15 14 of Z (, 1, or of Z + (, p, is in NP or in conp. h H The remaining open questions eal with a reformulation in term of egree sequence of hypergraphs. The question is presente within the context of H q + (, p but coul be aapte to H q (, p. Each subset H {0, 1} can be associate to a hypergraph with groun set []. The vector h is calle the egree sequence of H, an the convex hull of the egree sequences of all hypergraphs with groun set [] is calle the hypergraph polytope D ; an thus D = H + (, 1. Consiering only k-uniform hypergraphs; that is, subsets H {0, 1} where all vectors in H have k nonzero entries, one obtains the k-uniform hypergraph polytope D (k as the convex hull of the egree sequences of all k-uniform hypergraphs. The k-uniform hypergraph polytope, in particular D (2 an D (3, have been extensively stuie, see [5, 10, 17, 20] an references therein. A natural question raise in the literature asks for suitable necessary an sufficient conition to check whether a vector h D (k Z is the egree sequence of some k-uniform hypergraph. A trivial necessary conition is that the sum of the coorinates of h is a multiple of k. For k = 2; that is for graphs, the celebrate Er ós-gallai Theorem [10] shows that the trivial necessary conition is also sufficient. For k = 3; that is for 3-uniform hypergraphs, the question was raise by Klivans an Reiner [17]. Liu [18] exhibite counterexamples by constructing holes for 16; that is, vectors h in D (3 Z such that the sum of the coorinates of h is a multiple of 3, but h is not the egree sequence of a 3-uniform hypergraph. As there is no trivial congruence necessary conition, we call a vector in H + q (, p Z a hole if it cannot be written as the sum of a subset of the generators of H + q (, p. While the answer to the question Do H + q (, p have holes? is likely yes for most p, q,, it woul be interesting to explicitly fin such holes an better unerstan them. A natural follow-up question, provie there are holes, is For given fixe positive integers p an q, what is the complexity of eciing if a given vector h H + q (, p Z is a hole, an if not, of writing h as the sum of a subset of generators of H + q (, p?". As note in the proof of Proposition 4.1, there are polynomially many generators for fixe integer p an q. Thus, the above follow-up question is in conp as, if h is not a hole, it is possible to write h as a sum of a subset of generators H + q (, p. The last question is thus Is this problem conp-complete?". As for the linear optimization relate questions, the hole relate questions seem to be significantly harer for q =. In particular, for (q,, p = (,, 1, the questions investigates the holes of D. 5 Proofs for Sections 2.2 an 3 Let G q (, p, respectively G + q (, p, enote the generators of Z q (, p, respectively Z + q (, p. Recall that σ q (, p, respectively σ + q (, p, enotes the sum of the generators of Z q (, p,

16 15 respectively Z + q (, p. 5.1 Proof for Section Proof of item (i of Property 2.1 Proof. Note that if the set G of generators of a zonotope Z is is invariant uner coorinate permutation or sign flip, then the same hols for Z. Let π enote a permutation or a sign flip, an consier a signe sum ɛ g g. Then, π( ɛ g g = ɛ g π(g is also a signe sum g G g G g G of generators since G is permutation an sign flip invariant. In other wors, the set of all signe sums is invariant uner permutations an sign flips, an thus the same hols for the convex hull Z of all signe sums. Let J q (, p be the set of all g for g G q (, p. The zonotope Z q (, p generate by G q (, p J q (, p is the image of Z q (, p by a homothety of factor 2, an thus shares the same symmetry group. One can check that the set of generators of Z q (, p is invariant uner coorinate permutation or sign flip, thus the same hols for Z q (, p, an consequently hols for Z q (, p Proof of item (ii of Property 2.1 Proof. Consier the minimization problem {min c T x : x H q (, p} or, equivalently, min c T x over all integer value points of H q (, p. Set c = (! x, ( 1! x 1,..., x where x = (2p Assuming that x is not the origin, let x i0 enotes the first nonzero coorinate of x. Note that x i0 1 by efinition of G q (, p, an x i x. Thus, c T x ( + 1 i 0! x +1 i 0 x ( + 1 i! x +1 i > 0. In other wors, the origin is i 0 <i the unique minimizer of a linear optimization instance over H q (, p; that is, the origin is a vertex of H q (, p. As Z q (, p = 2H q (, p σ q (, p, the point σ q (, p is a vertex of Z q (, p. By item (i of Proposition 2.1, the point σ q (, p is a vertex of Z q (, p, an thus (σ q (, p + σ q (, p/2 is a vertex of H q (, p Proof of item (iii of Property 2.1 Proof. We first show that the coorinates of the vertex σ q (, p are o. As note in the proof f item (iii of Property 2.3, the i-th coorinate of σ q (, p is equal to the first coorinate of σ q ( i + 1, p. Thus, it is enough to show that the first coorinate of σ q (, p is o. Except for the first unit vector (1, 0,..., 0, any generator g of Z q (, p with nonzero first coorinate can be paire with the generator ḡ where ḡ 1 = g 1 an ḡ i = g i for i 1. Thus, the sum of the first coorinates of the generators of Z q (, p, excluing the the first unit vector, is even. Hence, the first coorinate of σ q (, p is o, an thus all the coorinates of σ q (, p are o. Consier a vertex v = ɛ(gg of g G q(,p Z q (, p. Since flipping the sign of an ɛ(g oes not change the parity of a coorinate of v, the coorinates of v have the same parity as the ones of σ q (, p; i.e. are o. In particular,

17 16 the coorinates of a vertex of Z q (, p are nonzero an item (i of Proposition 2.1 implies that the number of vertices of Z q (, p is a multiple of Proof of items (iv an (v of Property 2.1 Proof. Let Z be a zonotope generate by integer-value generators m j : j = 1,..., m(z. Then, Z is, up to translation, a lattice (, k-polytope with k max. Item m j i i=1,..., 1 j m(z (i of Property 2.1 implies that the integer range of its coorinates is inepenent of the chosen coorinate. The same hols for H q (, p, an, thus to etermine the integer range of H q (, p, it is enough to consier the first coorinates of its generators. Since the origin is a vertex of H q (, p an the first coorinate of its generator is nonnegative, the integer range of H q (, p is the sum of the first coorinates of its generators. For item (v, recall that the iameter of a zonotope is at most the number of its generators, an this inequality is satisfie with equality if no pair of generators are linearly epenent which is the case for Z q (, p an Z + q (, p Proof of Property 2.2 Proof. Consier a generator g G + q (, p an a coorinate permutation π. Since π(g G + q (, p, π(z + q (, p = π( [ 1, 1]G + q (, p = [ 1, 1]π(G + q (, p = [ 1, 1]G + q (, p = Z + q (, p. As in the proof of item (ii of Property 2.1, one can check that the origin is the unique minimizer of {min c T x : x H q (, p} with c = (1, 1,..., 1. Thus, the origin is a vertex of H + q (, p. As Z + q (, p = 2H + q (, p σ q (, p, the point σ q (, p is a vertex of Z + q (, p. Since Z + q (, p is invariant uner the symmetries inuce by coorinate permutations, σ q (, p is a vertex of Z + q (, p, an thus (σ q (, p + σ q (, p/2 is a vertex of H + q (, p Proof of items (i an (ii of Property 2.3 Proof. Given a canonical vertex v of Z q (, p, let c be a vector such that v is the unique maximizer of {max c T x : x Z q (, p}. Up to infinitesimal perturbations, we can assume that the coorinates of c are pairwise istinct an nonzero. Note that each coorinate c i of c is positive as otherwise flipping the sign of v i > 0 woul yiel a point in Z q (, p with higher objective value than v. Assume that c i < c j for some i < j. Then, v i = v j as otherwise permuting v i an v j woul yiel a point in Z q (, p with higher objective value than v. Let π ij (c be obtaine by permuting c i an c j. Then, one can check that v is the unique maximizer of {max π ij (c T x : x Z q (, p}. Assume, by contraiction, that v Z q (, p satisfies π ij (c T v π ij (c T v. Then, c T π ij (v = π ij (c T v π ij (c T v = c T v which implies π ij (v = v, an hence v = v, since v is the unique maximizer of {max c T x : x Z q (, p}. Thus, successive appropriate permutations yiel a vector π(c with π(c 1 > > π(c > 0 such that v is the unique maximizer of {max c T x : x Z q (, p}. For item (ii, one can check that σ 1 (, 2 = (2 1, 2 3,..., 1 is the unique maximizer of {max c T x : x Z 1 (2, p} for any c satisfying c 1 > > c > 0. Thus, by item (i of

18 17 Property 2.3, σ 1 (, 2 is the unique canonical vertex of Z 1 (, 2 an the vertices of Z 1 (, 2 are the 2! coorinate permutations an sign flips of σ 1 (, Proof of item (iii of Property 2.3 Proof. We first note that the i-th coorinate of σ q (, p is equal to the first coorinate of σ q ( i + 1, p. The statement trivially hols for i = 1. For i > 1, consier a generator g of Z q (, p with g i 0 an g i0 > 0 for some i 0 < i, then g can be paire with the generator ḡ where g i = ḡ i an g i0 = ḡ i0. Thus, the sum of all the i-th coorinates of the generators of Z q (, p is equal to the sum of the generators of Z q (, p such that the first i 1 coorinates are zero. In other wors, the i-th coorinate of σ q (, p is equal to the first coorinate of σ q ( i+1, p. Then, note that the first coorinate of σ q ( i+1, p, which is the gri embeing size of H q ( i+1, p, is strictly ecreasing with i increasing. Thus, the action of the symmetry group of Z q (, p on σ q (, p generates 2! istinct vertices of Z q (, p. For instance, one can check the i-th coorinate of σ (, 1 is 3 i Proof of item (iv of Property 2.3 Proof. The statement trivially hols for = 1. For 2, we show by inuction that the vertices of Z (, + 1 inclue σ( satisfying 0 = σ 1 ( < < σ ( = 2 1. The base case hols for = 2 as σ(2 = (0, 2 is a vertex of Z (2, + 1. Assume such a vertex σ( exists, an thus σ( = ɛ(gg for some ɛ(g an σ( is the unique maximizer g G + (,1 of {max c( T x : x Z (, + 1} for some c(. The generators of Z ( + + 1, 1 consist of the 2 1 vectors (g, 0 obtaine by appening 0 to a generator of Z (, + 1, the 2 1 vectors (g, 1 obtaine by appening 1, an the unit vector e +1. Consier the point s( + 1 = e +1 + (g, 1 ɛ(g(g, 0 = (2 1,..., 2 1, 2 (σ(, 0; that g G + (,1 g G + (,1 is, s( + 1 = (2 1 σ 1 (,..., 2 1 σ 1 (, 0, 2. Thus, the coorinates of s( + 1 are pairwise istinct an a suitable permutation of s( + 1 yiels a point σ( + 1 satisfying 0 = σ 1 ( + 1 < < σ +1 ( + 1 = 2. To show that σ( + 1 is a vertex of Z ( + + 1, 1, one can check that σ( + 1 is the unique maximizer of {max c(+1 T x : x Z (+1, + 1} where c( + 1 = ( c(, c +1 for sufficiently large c +1. Thus, for 2, a point σ( satisfying 0 = σ 1 ( < < σ ( = 2 1 is a vertex of Z q + (, p. Zonotopes being centrally symmetric, σ( is a vertex of Z q + (, p an the same hols for the istinct 2! permutations of ±σ(. 5.2 Proof for Section Proof of Property 3.1 Proof. One can check that the generators of H 1 (, 2 consist of ( ( 1 unity vectors an 2 2 vectors {..., 1,..., ±1,... }. Thus, the iameter of H 1 (, 2 is ( ( = 2. Similarly, one can check that the sum of the first coorinates of the generators of H 1 (, 2 is

19 Note that H 1 (, 2 is the permutaheron of type B. Then, one can check that, in aition to the previously etermine generators of H 1 (, 2, the generators of H 1 (, 3 consist of 2 ( ( 2 vectors {..., 1,..., ±2,... }, 2 ( 2 vectors {..., 2,..., ±1,... }, an 4 3 vectors {..., 1,..., ±1,..., ±1,... }. Thus, the iameter of H 1 (, 3 is ( ( ( = ( + 2(2 1/3. Similarly, one can check that the sum of the first coorinates of the generators of H 1 (, 3 is ( ( ( = Furthermore, one can check that, in aition to the previously etermine generators of H 1 (, 3, the generators of H 1 (, 4 consist of 2 ( ( 2 vectors {..., 1,..., ±3,... }, 2 ( 2 vectors {..., 3,..., ±1,... }, 4 3 vectors {..., 1,..., ±1,..., ±2,... }, 4 ( ( 3 vectors {..., 1,..., ±2,..., ±1,... }, 4 3 vectors {..., 2,..., ±1,..., ±1,... }, an 8 ( 4 vectors {..., 1,..., ±1,..., ±1,..., ±1,... }. Thus, the iameter of H 1 (, 4 is ( ( ( ( = ( /3. Similarly, one can check that the sum of the first coorinates of the generators of H 1 (, 4 is ( ( ( ( Finally, item (v correspons to Proposition j Proof of Property 3.2 Proof. One can check that the generators of H 2 (, 2 consist of ( ( 1 unity vectors, 2 2 vectors {..., 1,..., ±1,... }, 4 ( ( 3 vectors {..., 1,..., ±1,..., ±1,... }, an 8 4 vectors {..., 1,..., ±1,..., ±1,..., ±1,... }. Thus, the iameter of H 2 (, 2 is 2 j( j+1. Similarly, one can check that the sum of the first coorinates of the generators of H 2 (, 2 is 2 j( 1 j. 0 j Proof of Property 3.3 Proof. One can check that H (, 1 has (3 1/2 generators consisting of all { 1, 0, 1}- value vectors which first nonzero coorinate is positive. Out of the 5 { 2, 1, 0, 1, 2}- value vectors, 3 are { 2, 0, 2}-value. Thus, keeping the ones which first nonzero coorinate is positive, H (, 2 has (5 3 /2 generators. Similarly, one can check that the sum of the first coorinates of the generators of H (, 2 is The generators (i, j of H (2, p such that (i, j 1 are (1, 0, (0, 1, (1, 1 an (1, 1. For a given i > 1, there are 2φ(i generators (i, j such that (i, j > 1 an j < i. Thus, there are 4 φ(j generators (i, j such that (i, j > 1. Thus, the iameter of H (2, p is 2 j p 4 φ(j. 1 j p Proof of Property 3.4 Proof. One can check that the generators of H 1 + (, 2 consist of ( ( 1 unity vectors an 2 vectors {..., 1,..., 1,... }. Thus, the iameter of H 1 + (, 2 is ( ( 1 + ( 2 = Similarly, one can check that the sum of the first coorinates of the generators of H 1 + (, 2 is. Note that H 1 + (2, p is the Minkowski sum of the permutaheron with the {0, 1} -cube. One can check that, in aition to the previously etermine generators of H 1 + (2, p,

20 19 the generators of H 1 + (, 3 consist of ( ( 3 vectors {..., 1,..., 1,..., 1,..., }, 2 vectors {..., 1,..., 2,... }, an ( 2 vectors {..., 2,..., 1,... }. Thus H + 1 (, 3 has ( ( ( generators. Similarly, one can check that the sum of the first coorinates of the generators of H 1 + (, 3 is ( ( ( Out of the generators of H1 (2, p, φ(j have a 2 j p negative coorinate. Thus, the iameter of H 1 + (2, p is 1 + φ(j. Similarly, one 1 j p can check that the sum of the first coorinates of the generators of H 1 + (2, p is 1 + jφ(j/2. 2 j p Proof of Property 3.5 Proof. One can check that the generators of H 2 + (, 2 consist of ( i vectors with exactly i ones for i = 1, 2, 3, an 4. Thus, the iameter of H 2 + (, 2 is ( ( Similarly, one can check that the sum of the first coorinates of the generators of H 2 + (, 2 is ( ( Proof of Property 3.6 Proof. One can check that H (, + 1 has 2 1 generators consisting of all {0, 1}-value vectors except the origin. Thus, the iameter of H (, + 1 is 2 1. Similarly, one can check that the sum of the first coorinates of the generators of H (, + 1 is 2 1. Out of the 3 {0, 1, 2}-value vectors, 2 are {0, 2}-value. Thus, the iameter of H (, + 2 is 3 2. Similarly, one can check that the sum of the first coorinates of the generators of H (, + 2 is 3 2. The generators (i, j of H (2, + p such that (i, j 1 are (1, 0, (0, 1, an (1, 1. For a given i > 1, there are φ(i generators (i, j such that (i, j > 1 an j < i. Thus, there are 2 φ(j generators (i, j such that (i, j > 1. Thus, the iameter 2 j p of H (2, + p is φ(j. 1 j p Acknowlegment. The authors thank the anonymous referees, Johanne Cohen, Nathann Cohen, Komei Fukua, an Alain Virmaux for valuable comments an for informing us of reference [28], Emo Welzl an Günter Ziegler for helping us access Thorsten Thiele s Diplomarbeit, Dmitrii Pasechnik for pointing out reference [26], an Vincent Pilau for pointing out that Z 1 (, 2 is the permutaheron of type B. This work was partially supporte by the Natural Sciences an Engineering Research Council of Canaa Discovery Grant program (RGPIN , by the Digiteo Chair C&O program, an by the Dresner Chair at the Technion. References [1] Dragan Acketa an Joviša Žunić. On the maximal number of eges of convex igital polygons inclue into an m m-gri. Journal of Combinatorial Theory A, 69: , 1995.

21 20 [2] Xavier Allamigeon, Pascal Benchimol, Stéphane Gaubert, an Michael Joswig. Long an wining central paths. arxiv: , [3] Antal Balog an Imre Bárány. On the convex hull of the integer points in a isc. In Proceeings of the Seventh Annual Symposium on Computational Geometry, pages , [4] Nicolas Bonifas, Marco Di Summa, Frierich Eisenbran, Nicolai Hähnle, an Martin Niemeier. On sub-eterminants an the iameter of polyhera. Discrete an Computational Geometry, 52: , [5] Charles Colbourn, William Kocay, an Douglas Stinson. Some NP-complete problems for hypergraph egree sequences. Discrete Applie Mathematics, 14: , [6] Alberto Del Pia an Carla Michini. On the iameter of lattice polytopes. Discrete an Computational Geometry, 55: , [7] Antoine Deza, George Manoussakis, an Shmuel Onn. Primitive zonotopes. Discrete an Computational Geometry, (to appear. [8] Antoine Deza an Lionel Pournin. Improve bouns on the iameter of lattice polytopes. arxiv: , [9] Davi Eppstein. Zonohera an zonotopes. Mathematica in Eucation an Research, 5:15 21, [10] Paul Erös an Tibor Gallai. Graphs with prescribe egrees of vertices (in Hungarian. Matematikai Lopak, 11: , [11] Komei Fukua. Lecture notes: Polyheral computation. ethz.ch/personal/fukuak/lect/pclect/notes2015/. [12] Martin Grötschel, Laszlo Lovász, an Alexaner Schrijver. Geometric algorithms an combinatorial optimization. Springer, [13] Branko Grünbaum. Convex Polytopes. Grauate Texts in Mathematics. Springer, [14] James Humphreys. Reflection groups an Coxeter groups. Cambrige Stuies in Avance Mathematics. Cambrige University Press, [15] Gil Kalai an Daniel Kleitman. A quasi-polynomial boun for the iameter of graphs of polyhera. Bulletin of the American Mathematical Society, 26: , [16] Peter Kleinschmit an Shmuel Onn. On the iameter of convex polytopes. Discrete Mathematics, 102:75 77, 1992.

22 21 [17] Caroline Klivans an Vic Reiner. Shifte set families, egree sequences, an plethysm. Electronic Journal of Combinatorics, 15 (1, [18] Ricky Liu. Nonconvexity of the set of hypergraph egree sequences. Electronic Journal of Combinatorics, 20 (1, [19] Michal Melame an Shmuel Onn. Convex integer optimization by constantly many linear counterparts. Linear Algebra an its Applications, 447:88 109, [20] N.L. Bhanu Murthy an Murali K. Srinivasan. The polytope of egree sequences of hypergraphs. Linear Algebra an its Applications, 350: , [21] Dennis Naef. The Hirsch conjecture is true for (0, 1-polytopes. Mathematical Programming, 45: , [22] Shmuel Onn. Nonlinear Discrete Optimization. Zurich Lectures in Avance Mathematics. European Mathematical Society, [23] Shmuel Onn an Uriel G. Rothblum. Convex combinatorial optimization. Discrete an Computational Geometry, 32: , [24] Francisco Santos. A counterexample to the Hirsch conjecture. Annals of Mathematics, 176: , [25] Neil Sloane (eitor. The on-line encyclopeia of integer sequences. org. [26] Ivan Soprunov an Jenya Soprunova. Eventual quasi-linearity of the Minkowski length. European Journal of Combinatorics, 58: , [27] Noriyoshi Sukegawa. Improving bouns on the iameter of a polyheron in high imensions. arxiv: , [28] Torsten Thiele. Extremalprobleme für Punktmengen. Diplomarbeit, Freie Universität Berlin, [29] Michael To. An improve Kalai-Kleitman boun for the iameter of a polyheron. SIAM Journal on Discrete Mathematics, 28: , [30] Günter Ziegler. Lectures on Polytopes. Grauate Texts in Mathematics. Springer, Antoine Deza, Avance Optimization Laboratory, Faculty of Engineering McMaster University, Hamilton, Ontario, Canaa. eza@mcmaster.ca

23 22 George Manoussakis, Laboratoire e Recherche en Informatique Université e Paris Su, Orsay, France. george@lri.fr Shmuel Onn, Operations Research, Davison faculty of IE & M Technion Israel Institute of Technology, Haifa, Israel. onn@ie.technion.ac.il