The height of minimal Hilbert bases

Transcription

1 1 The height of minimal Hilbet bases Matin Henk and Robet Weismantel Abstact Fo an integal polyhedal cone C = pos{a 1,..., a m, a i Z d, a subset BC) C Z d is called a minimal Hilbet basis of C iff i) each element of C Z d can be witten as a non-negative integal combination of elements of BC) and ii) BC) has minimal cadinality with espect to all subsets of C Z d fo which i) holds. We give a tight bound fo the so-called height of an element of the basis which impoves on fome esults. 1 Intoduction Thoughout the pape R d denotes the d-dimensional Euclidean space, and pos S the positive hull of a subset S R d. The cadinality of a finite) subset S R d is denoted by #S and the i-th unit vecto is epesented by e i. A cone C R d is a set with the popeties that x + y C if x, y C and λx C if x C, λ 0. A cone C is called pointed if the set C\{0 is stictly contained in an open halfspace, i.e., thee exists c R d such that c T x < 0 fo all x C\{0. If C = pos{a 1,..., a m with vectos a i R d, 1 i m, then C is called a polyhedal cone o a finitely geneated cone. Hee we ae studying integal polyhedal cones C R d, i.e, thee exist vectos a i Z d \{0 fo 1 i m such that C = pos{a 1,..., a m, o equivalently, C = {x R d : Ax 0 fo an appopiate matix A Z n d. Fom Godan s lemma cf. [1], [2]) we know that fo evey integal polyhedal cone C thee exists a set BC) C Z d such that 1. each z C Z d can be expessed as a non-negative integal combination of elements in BC), i.e., z = b BC) z b b, z b N. 2. BC) has minimal cadinality with espect to all subsets of C Z d fo which 1) holds. BC) is called a minimal Hilbet basis of C. Fo shot we just say basis of C. If C is a pointed cone then BC) is uniquely detemined cf. [3], [2]), { BC) = b C Z d \{0 : b is not the sum of two othe 1.1) vectos in C Z d \{ Mathematics Subject Classification. Pimay 52B20, 90C21; Seconday 11P21. Key wods and Phases. Hilbet bases, ational polyhedal cones, intege pogamming.

2 2 This implies that the basis BC) is contained in the zonotope geneated by a 1,..., a m. pecisely, we have Moe BC) {a 1,..., a m { a C Z d \{0 : a = m λ i a i, 0 λ i < 1, 1 i m. 1.2) We want to emak that minimal) Hilbet bases occu unde many diffeent names in vaious fields of mathematics such as intege pogamming cf. [2], [4], [5], [6]) o in the context of special desingulaizations of toic vaieties cf. [7], [8], [9],[10]). Howeve, vey little is known about the geometical stuctue of the basis elements. Hee we focus on the height of the Hilbet basis. Definition 1. Let C = pos{a 1,..., a m, a i Z d, be a pointed cone. Fo b BC) the numbe { m m h C b) := min λ i : b = λ i a i, λ i 0, 1 i m is called the height of the basis element b. It is staightfowad to see that fo dimension 2 the height is not geate than 1 and fom Caatheodoy s theoem and 1.2) one easily deives the bound h C b) < d, b BC), C R d. Indeed, it was poven by Ewald & Wessels [11] that h C b) < d 1, b BC), d 3, is an asymptotically tight uppe bound fo the height see also [12]). Hee we shapen the bound in the following way. Theoem 1. Let C = pos{a 1,..., a m, a i Z d, be a d-dimensional pointed cone. Fo b BC) one has h C b) d 1) deta i, 1,..., a i d) whee {a i 1,..., a i d {a 1,..., a m is a subset of d linealy independent integal points such that b pos{a i 1,..., a i d. This bound is tight fo vaious families of cones. Fo example, let N\{0 and let cf. [11]) { d 1 C d = pos e 1,..., e d 1, e d + e i. The point b = 1,..., 1) T is an element of the basis with h C b) = d 1) 1 2 Poof of Theoem 1 The poof is pepaed by the next two simple lemmas. + 1 = d 1 dete 1,..., e d 1, e d + d 1 ei ).

3 3 Lemma 1. Let p, N such that 1 p 1. We define Mp, ) = {j {0,..., 1 : j p) mod p. Then #Mp, ) = p + gcdp, ). Poof. Obviously, if gcdp, ) = 1 then {j p) mod : 0 j 1 = {0,..., 1 and the statement is tue. Hence #Mp/q, /q) = p/q + 1 whee q = gcdp, ). Since j + i ) p mod = q j p q q mod ), 0 j 1, 0 i q 1, q q we get #Mp, ) = q #Mp/q, /q) = p + gcdp, ). The next lemma is quite obvious and can easily be poved by induction on the numbe n. Lemma 2. Let m, n be positive integes and let N i {1,..., n fo 1 i m. If m #N i m 1) n + k, k {1,..., n, then # m N i) k. We ae now eady fo the poof of Theoem 1. Poof of Theoem 1. Let b BC) and w.l.o.g. let {a 1,..., a d be a subset of the geneatos a 1,..., a m of the cone C such that a 1,..., a d ae linealy independent and b pos{a 1,..., a d. Let Λ Z d be the lattice geneated by {a 1,..., a d, b, detλ) its deteminant and = deta 1,..., a d ) / detλ) N be the index of the sublattice geneated by a 1,..., a d w..t. Λ cf. [13]). In the following we show detλ) h C b) d 1) ) deta 1,..., a d ) = d 1), 2.1) which is a slightly stonge inequality as posed in Theoem 1. To this end let b / {a 1,..., a d. Futhemoe, since b is also contained in the minimal Hilbet basis of the cone pos{a 1,..., a d we may assume by 1.2) that b = d λ ia i with 0 λ i < 1. It is quite easy to see that the coefficients λ i have a epesentation as λ i = p i, p i {0,..., 1, 1 i d, with gcdp 1,..., p d, ) = 1 and that { ) j pi ) mod a i : 1 j 1 C Z d \{0. 2.2) Now, by definition we have h C b) d p i/) and thus it suffices to show d )/ cf. 2.1)). Assume the contay, i.e., p i d 1) p i d 1) 1) ) Then 3 and we show that b can be witten as the sum of two elements contained in the set on the left hand side of 2.2). This contadicts 1.1). Fo 1 i d let Mp i, ) = {j {0,..., 1 : jp i mod p i.

4 4 Lemma 1 yields the bound #Mp i, ) p i + 1 and by 2.3) we get #Mp i, ) d 1) + 3. Lemma 2 says that the intesection d Mp i, ) contains an element k {2,..., 1, say. By the definition of the sets Mp i, ) we have ) ) p i = k p i ) mod + 1 k) p i ) mod, 1 i d, and we get the desied contadiction b = ) k pi ) mod a i + 1 k) pi ) mod ) a i. 3 Some consequences of Theoem 1 Theoem 1 may be used to deive the following lowe bound. Coollay 1. Let C = pos{a 1,..., a d be a pointed cone such that a 1,..., a d Z d ae linealy independent. If BC) = {a 1,..., a d {z Z d : z = d λ ia i, 0 λ i < 1 then fo b BC) contained in the inteio of C one has h C b) 1 + deta 1,..., a d ). Poof. Since b is contained in the inteio of C the lattice point b = d ai b is contained in the half open paallelepiped geneated by a 1,..., a d. Hence, by assumption b BC) and Theoem 1 yields h C b) = d h C b) 1 + )/ deta 1,..., a d ). The next coollay shows an application of Theoem 1 in intege pogamming. Coollay 2. Let A Z m d with all subdeteminants at most α in absolute value and let b Z m, c Z d be given vectos. If z is a feasible, non-optimal solution of the pogam max{c T z : Az b, z Z d, then thee exists a feasible solution ẑ such that c T ẑ > c T z and whee denotes the maximum nom. ẑ z d 1)α d d/2 α d 2, Poof. Let z be a feasible solution with c T z > c T z. We split the system Ax b into subsystems A 1 x b 1, A 2 x b 2 such that A 1 z A 1 z and A 2 z A 2 z. Let C be the cone C = {x R d : A 1 x 0, A 2 x 0

5 5 and w 1,..., w n Z d such that C = pos{w 1,..., w n. Using Came s ule we obtain that w j α, 1 j n. Since z z C thee exist l d linealy independent vectos w i 1,..., w i l such that z z C = pos{w i 1,..., w i l. It follows that z z = k n i b i, n i N\{0 fo some b 1,..., b k BC). It is easy to see that z + b i, i {1,..., k, is a feasible solution. On account of the condition c T z > c T z we may assume that c T z + b 1 ) > c T z. We define ẑ := z + b 1 and wite b 1 as l b 1 = λ j w i j j=1 with λ 1,..., λ l 0. Applying Theoem 1 to the l-dimensional cone C togethe with the Hadamad inequality gives l ) ẑ z α λ j l 2 α l 1) w i. 1 w i l ) As w i j d 1/2 α, we obtain j=1 ẑ z l 1)α l 2 d l/2 d 1)α αl 1 d d/2 α d 1, whee the last inequality can be veified with elementay algeba. We emak that the bound of Coollay 2 stengthens the bound of dα given in [14]. We ae gateful to Les Totte j. and Günte M. Ziegle fo helpful discussions and comments. Refeences [1] P. Godan, Übe die Auflösung lineae Gleichungen mit eellen Coefficienten, Math. Ann ), [2] A. Schijve, Theoy of linea and intege pogamming, John Wiley and Sons, Chicheste, [3] J.G. van de Coput, Übe Systeme von linea-homogenen Gleichungen und Ungleichungen, Poceedings Koninklijke Akademie van Wetenschappen te Amstedam ), [4] A. Sebö, Hilbet bases, Caatheodoy s Theoem and combinatoial optimization, Poc. of the IPCO confeence, Wateloo, Canada, 1990, [5] R. Ubaniak, R. Weismantel, and G.M. Ziegle, A vaiant of Buchbege s algoithm fo intege pogamming, SIAM J. Discete Math., to appea. [6] R. Weismantel, Hilbet bases and the facets of special knapsack polytopes, Math. Ope. Res. 214) 1996), [7] C. Bouvie and G. Gonzalez-Spingbeg, G-désingulaisations de vaiétés toiques, C. R. Acad.Sci. Pais, t.315, Séie I, 1992),

6 6 [8] D. Dais, M. Henk, and G.M. Ziegle, On the existence of cepant esolutions of Goenstein abelian quotient singulaities in dimensions 4, in pepaation 1996). [9] G. Ewald, Combinatoial Convexity and Algebaic Geomety, Gaduate Texts in Mathematics Vol. 168, Spinge, Belin, [10] T. Oda, Convex bodies and algebaic geomety, Spinge, New-Yok, [11] G. Ewald and U. Wessels, On the ampleness of invetible sheaves in complete pojective toic vaieties, Result. Math ), [12] J. Liu, L.E. Totte, J., and G.M. Ziegle, On the Height of the minimal Hilbet basis, Result. Math ), [13] P.M. Gube and C.G. Lekkekeke, Geomety of numbes, 2nd ed., Noth-Holland, Amstedam, [14] W. Cook, A.M.H. Geads, A. Schijve, and É. Tados, Sensitivity theoems in intege pogamming poblems, Math. Pogamming ), Matin Henk, Robet Weismantel Konad-Zuse-Zentum fü Infomationstechnik ZIB) Belin Takustaße 7 D Belin-Dahlem, Gemany henk@zib.de, weismantel@zib.de Eingegangen am 30. Apil 1997.