FPTAS for optimizing polynomials over the mixed-integer points of polytopes in fixed dimension

Transcription

1 Matheatical Prograing anuscript No. (will be inserted by the editor) Jesús A. De Loera Rayond Heecke Matthias Köppe Robert Weisantel FPTAS for optiizing polynoials over the ixed-integer points of polytopes in fixed diension Revision: 1.50 Date: 2007/05/22 22:00:36 Abstract We show the existence of a fully polynoial-tie approxiation schee (FPTAS) for the proble of axiizing a non-negative polynoial over ixedinteger sets in convex polytopes, when the nuber of variables is fixed. Moreover, using a weaker notion of approxiation, we show the existence of a fully polynoial-tie approxiation schee for the proble of axiizing or iniizing an arbitrary polynoial over ixed-integer sets in convex polytopes, when the nuber of variables is fixed. Keywords Mixed-integer nonlinear prograing Integer prograing in fixed diension Coputational coplexity Approxiation algoriths FPTAS Matheatics Subject Classification (2000) 90C11 90C30 90C60 90C57 A conference version of this article, containing a part of the results presented here, appeared in Proceedings of the 17th Annual ACM-SIAM Syposiu on Discrete Algoriths, Miai, FL, January 22 24, 2006, pp The first author gratefully acknowledges support fro NSF grant DMS , a 2003 UC-Davis Chancellor s fellow award, the Alexander von Huboldt foundation, and IMO Magdeburg. The reaining authors were supported by the European TMR network ADONET J.A. De Loera University of California, Dept. of Matheatics, Davis CA 95616, USA E-ail: deloera@ath.ucdavis.edu R. Heecke Otto-von-Guericke-Universität Magdeburg, FMA/IMO, Universitätsplatz 2, Magdeburg, Gerany E-ail: heecke@io.ath.uni-agdeburg.de M. Köppe Otto-von-Guericke-Universität Magdeburg, FMA/IMO, Universitätsplatz 2, Magdeburg, Gerany E-ail: koeppe@io.ath.uni-agdeburg.de R. Weisantel Otto-von-Guericke-Universität Magdeburg, FMA/IMO, Universitätsplatz 2, Magdeburg, Gerany E-ail: weisant@io.ath.uni-agdeburg.de

2 2 J.A. De Loera, R. Heecke, M. Köppe, R. Weisantel 1 Introduction A well-known result by H.W. Lenstra Jr. states that integer linear prograing probles with a fixed nuber of variables can be solved in polynoial tie on the input size [12]. Likewise, ixed integer linear prograing probles with a fixed nuber of integer variables can be solved in polynoial tie. It is a natural question to ask what is the coputational coplexity, when the nuber of variables (or the nuber of integer variables) is fixed, of the non-linear ixed integer proble ax f (x 1,...,x d1,z 1,...,z d2 ) (1a) s.t. Ax + Bz b (1b) x i R for i = 1,...,d 1, (1c) z i Z for i = 1,...,d 2, (1d) where f is a polynoial function of axiu total degree D with rational coefficients, and A Z p d 1, B Z p d 2, b Z p. We are interested in general polynoial objective functions f without any convexity assuptions. Throughout the paper we assue that the inequality syste Ax + Bz b describes a convex polytope, i.e., a bounded polyhedron, which we denote by P. The reason for this restriction are fundaental noncoputability results for probles involving polynoials and integer variables. Indeed, when we perit unbounded feasible regions, there cannot exist any algorith to decide whether there exists a feasible solution to (1) with f (x,z) α (for a prescribed bound α), ruling out the existence of an optiization algorith or any approxiation schee. This is due to the negative answer to Hilbert s tenth proble by Matiyasevich [13,14]. Due to Jones strengthening of this negative result [10], there also cannot exist any such algorith for the cases of unbounded feasible regions for any fixed nuber of integer variables d For the purpose of coplexity analysis, we assue that the data A, B, and b are given by the binary encoding schee, and that the objective function f is given as a list of onoials, where the coefficients are encoded using the binary encoding schee and the exponent vectors are encoded using the unary encoding schee. In other words, the running ties are peritted to grow polynoially not only in the binary encoding of all the proble data, but also in the axiu total degree D of the objective function f. It is well-known that pure continuous polynoial optiization over polytopes (d 2 = 0) in varying diension is NP-hard and that a fully polynoial tie approxiation schee (FPTAS) is not possible (unless P = NP). Indeed the ax-cut proble can be odeled as iniizing a quadratic for over the cube [ 1,1] d. Håstad [9] proved that the ax-cut proble cannot be approxiated to a ratio better than (unless P = NP). This excludes the possibility of a polynoial tie approxiation schee for (1) in varying diension, even when the nuber of integer variables is fixed. On the other hand, pure continuous polynoial optiization probles over polytopes (d 2 = 0) can be solved in polynoial tie when the diension d 1 is fixed. This follows fro a uch ore general result on the coputational coplexity of approxiating the solutions to general algebraic forulae over the real nubers by Renegar [19]; see also [16,17,18].

3 FPTAS for ixed-integer polynoial optiization in fixed diension 3 However, when we perit integer variables (d 2 > 0), it turns out that, even for fixed diension d 1 + d 2 = 2 and objective functions f of axiu total degree D = 4, proble (1) is an NP-hard proble [6]. Thus the best we can hope for, even when the nuber of both the continuous and the integer variables is fixed, is an approxiation result. This paper presents the best possible such result: Theore 1 (Fully polynoial-tie approxiation schee) Let the diension d = d 1 + d 2 be fixed. (a) There exists a fully polynoial tie approxiation schee (FPTAS) for the optiization proble (1) for all polynoial functions f (x 1,...,x d1,z 1,...,z d2 ) with rational coefficients that are non-negative on the feasible region (1b 1d). (b) Moreover, the restriction to non-negative polynoials is necessary, as there does not even exist a polynoial tie approxiation schee (PTAS) for the axiization of arbitrary polynoials over ixed-integer sets in polytopes, even for fixed diension d 2, unless P = NP. The proof of Theore 1 is presented in section 5. As we will see, Theore 1 is a non-trivial consequence of the existence of FPTAS for the proble of axiizing a non-negative polynoial with integer coefficients over the lattice points of a convex rational polytope. That such FPTAS indeed exist was recently settled in our paper [6]. The knowledge of paper [6] is not necessary to understand this paper but, for convenience of the reader, we include a short suary in section 2. Our arguents, however, are independent of which FPTAS is used in the integral case. Our ain approach is to use grid refineent in order to approxiate the ixedinteger optial value via auxiliary pure integer probles. One of the difficulties on constructing approxiations is the fact that not every sequence of grids whose widths converge to zero leads to a convergent sequence of optial solutions of grid optiization probles. This difficulty is addressed in section 3. In section 4 we develop techniques for bounding differences of polynoial function values. Section 5 contains the proof of Theore 1. Finally, in section 6, we study a different notion of approxiation. The usual definition of an FPTAS uses the notion of ε-approxiation that is coon when considering cobinatorial optiization probles, where the approxiation error is copared to the optial solution value, f (xε,z ε ) f (x ax,z ax ) ε f (xax,z ax ), (2) where (x ε,z ε ) denotes an approxiate solution and (x ax,z ax ) denotes a axiizer of the objective function. In section 6, we now copare the approxiation error to the range of the objective function on the feasible region, f (xε,z ε ) f (x ax,z ax ) ε f (xax,z ax ) f (x in,z in ), (3) where additionally (x in,z in ) denotes a iniizer of the objective function on the feasible region. This notion of approxiation was proposed by various authors [20,3,11]. It enables us to study objective functions that are not restricted to be non-negative on the feasible region. We reark that, when the objective function can take negative values on the feasible region, (3) is weaker than (2). Therefore Theore 1 (b) does not rule out the existence of an FPTAS with respect to this notion of approxiation. Indeed we prove:

4 4 J.A. De Loera, R. Heecke, M. Köppe, R. Weisantel Theore 2 (Fully polynoial-tie weak-approxiation schee) Let the diension d = d 1 + d 2 be fixed. Let f be an arbitrary polynoial function with rational coefficients and axiu total degree D, and let P R d be a rational convex polytope. (a) In tie polynoial in the input size and D, it is possible to decide whether f is constant on P ( R d 1 Z d ) 2. (b) In tie polynoial in the input size, D, and ε 1 it is possible to copute a solution (x ε,z ε ) P ( R d 1 Z d ) 2 with f (xε,z ε ) f (x ax,z ax ) ε f (xax,z ax ) f (x in,z in ). Notation. As usual, we denote by Q[x 1,...,x d1,z 1,...,z d2 ] the ring of ultivariate polynoials with rational coefficients. For writing ultivariate polynoials, we frequently use the ulti-exponent notation, z α = z α 1 1 zα d d. 2 An FPTAS for the integer case The first fully polynoial-tie approxiation schee for the integer case appeared in our paper [6]. It is based on Alexander Barvinok s theory for encoding all the lattice points of a polyhedron in ters of short rational functions [1,2]. The set P Z d is represented by a Laurent polynoial g P (z) = α P Z d zα. Fro Barvinok s theory this exponentially-large su of onoials g P (z) can be written as a polynoial-size su of rational functions (assuing the diension d is fixed) of the for: z g P (z) = u i E i i I d j=1 (1 zv i j), (4) where I is a polynoial-size indexing set, and where E i {1, 1} and u i,v i j Z d for all i and j. There is a polynoial-tie algorith for coputing this representation [1,2,5,7]. By sybolically applying differential operators to the representation (4), we can copute a short rational function representation of the Laurent polynoial g P, f (z) = α P Z d f (α)z α. (5) In fixed diension, the size of the expressions occuring in the sybolic calculation can be bounded polynoially: Lea 3 ([6], Lea 3.1) Let the diension d be fixed. Let g P (z) = α P Z d zα be the Barvinok representation of the generating function of P Z d. Let f Z[x 1,...,x d ] be a polynoial of axiu total degree D. We can copute, in tie polynoial in D and the input size, a Barvinok representation g P, f (z) for the generating function α P Z d f (α)zα. Now we present the algorith to obtain bounds U k,l k that reach the optiu. We ake use of the eleentary fact that, for a set S = {s 1,...,s r } of non-negative real nubers, k ax{s 1,...,s r } = li s r k s r k. (6)

5 FPTAS for ixed-integer polynoial optiization in fixed diension 5 Algorith 4 (Coputation of bounds) Input: A rational convex polytope P R d, a polynoial objective f Z[x 1,...,x d ] of axiu total degree D that is non-negative over P Z d. Output: A nondecreasing sequence of lower bounds L k, and a nonincreasing sequence of upper bounds U k, both reaching the axial function value f of f over P Z d in a finite nuber of steps. 1. Copute a short rational function expression for the generating function g P (z) = α P Z d zα. Using residue techniques, copute P Z d = g P (1) fro g P (z). 2. Fro the rational function g P (z) copute the rational function representation of g P, f k(z) of α P Z d f k (α)z α by Lea 3. Using residue techniques, copute k k L k := g P, f k(1)/g P, f 0(1) and U k := g P, f k(1). Theore 5 ([6], Lea 3.3 and Theore 1.1) Let the diension d be fixed. Let P R d be a rational convex polytope. Let f be a polynoial with integer coefficients and axiu total degree D that is non-negative on P Z d. (i) Algorith 4 coputes the bounds L k, U k in tie polynoial in k, the input size of P and f, and the total degree D. The bounds satisfy the following inequality: ( ) U k L k f k P Z d 1. (ii) For k = (1 + 1/ε)log( P Z d ) (a nuber bounded by a polynoial in the input size), L k is a (1 ε)-approxiation to the optial value f and it can be coputed in tie polynoial in the input size, the total degree D, and 1/ε. Siilarly, U k gives a (1 + ε)-approxiation to f. (iii) With the sae coplexity, by iterated bisection of P, we can also find a feasible solution x ε P Z d with f (xε ) f ε f. 3 Grid approxiation results An iportant step in the developent of an FPTAS for the ixed-integer optiization proble is the reduction of the ixed-integer proble (1) to an auxiliary optiization proble over a lattice 1 Zd 1 Z d 2. To this end, we consider the grid proble with grid size, ax f (x 1,...,x d1,z 1,...,z d2 ) s.t. Ax + Bz b x i 1 Z for i = 1,...,d 1, z i Z for i = 1,...,d 2. (7) We can solve this proble approxiately using the integer FPTAS (Theore 5):

6 6 J.A. De Loera, R. Heecke, M. Köppe, R. Weisantel Corollary 6 For fixed diension d = d 1 + d 2 there exists an algorith with running tie polynoial in log, the encoding length of f and of P, the axiu total degree D of f, and ε 1 for coputing a feasible solution (x ε,z ε ) P ( 1 Z d 1 Z d ) 2 to the grid proble (7) with an objective function f that is non-negative on the feasible region, with f (x ε,z ε ) (1 ε) f (x,z ), (8) where (x,z ) P ( 1 Z d 1 Z d 2 ) is an optial solution to (7). Proof We apply Theore 5 to the pure integer optiization proble: ax f ( x, z) s.t. A x + Bz b x i Z for i = 1,...,d 1, z i Z for i = 1,...,d 2, where f ( x,z) := D f ( 1 x,z) is a polynoial function with integer coefficients. Clearly the binary encoding length of the coefficients of f increases by at ost Dlog, copared to the coefficients of f. Likewise, the encoding length of the coefficients of B and b increases by at ost log. By Theore 1.1 of [6], there exists an algorith with running tie polynoial in the encoding length of f and of Ax+Bz b, the axiu total degree D, and ε 1 for coputing a feasible solution (x ε,z ε ) P ( 1 Z d 1 Z d ) 2 such that f (x ε,z ε ) (1 ε) f (x,z ), which iplies the estiate (8). One ight be tepted to think that for large-enough choice of, we iediately obtain an approxiation to the ixed-integer optiu with arbitrary precision. However, this is not true, as the following exaple deonstrates. Exaple 7 Consider the ixed-integer optiization proble ax s.t. 2z x z 2x z 2(1 x) x R 0, z {0,1}, whose feasible region consists of the point ( 2 1,1) and the segent {(x,0) : x [0,1]}. The unique optial solution to (10) is x = 2 1, z = 1. Now consider the sequence of grid approxiations of (10) where x 1 Z 0. For even, the unique optial solution to the grid approxiation is x = 2 1, z = 1. However, for odd, the unique optial solution is x = 0, z = 0. Thus the full sequence of the optial solutions to the grid approxiations does not converge since it has two liit points; see Figure 1. (9) (10) Even though taking the liit does not work, taking the upper liit does. More strongly, we can prove that it is possible to construct, in polynoial tie, a subsequence of finer and finer grids that contain a lattice point ( x δ,z ) that is arbitrarily close to the ixed-integer optiu (x,z ). This is the central stateent of this section and a basic building block of the approxiation result.

7 FPTAS for ixed-integer polynoial optiization in fixed diension 7 Z 1 f ( 1 2,1) = 1 1 Z R f (0,0) = R Fig. 1 A sequence of optial solutions to grid probles with two liit points, for even and for odd Theore 8 (Grid Approxiation) Let d 1 be fixed. Let P = {(x,z) R d 1+d 2 : Ax + Bz b}, where A Z p d 1, B Z p d 2. Let M R be given such that P {(x,z) R d 1+d 2 : x i M for i = 1,...,d 1 }. There exists a polynoial-tie algorith to copute a nuber such that for every (x,z ) P (R d 1 Z d 2) and δ > 0 the following property holds: Every lattice 1 Zd 1 for = k and k 2 δ d 1M contains a lattice point x δ such that ( x δ,z ) P ( 1 Z d 1 Z d ) 2 and x δ x δ. The geoetry of Theore 8 is illustrated in Figure 2. The notation x δ has been chosen to suggest that the coordinates of x have been rounded to obtain a nearby lattice point. The rounding ethod is provided by the next two leas; Theore 8 follows directly fro the. Lea 9 (Integral Scaling Lea) Let P = {(x,z) R d 1+d 2 : Ax + Bz b}, where A Z p d 1, B Z p d 2. For fixed d 1, there exists a polynoial tie algorith to copute a nuber Z >0 such that for every z Z d 2 the polytope P z = { x : (x,z) P } is integral, i.e., all vertices have integer coordinates. In particular, the nuber has an encoding length that is bounded by a polynoial in the encoding length of P. Proof Because the diension d 1 is fixed, there exist only polynoially any siplex bases of the inequality syste Ax b Bz, and they can be enuerated in polynoial tie. The deterinant of each siplex basis can be coputed in polynoial tie. Then can be chosen as the least coon ultiple of all these deterinants. Lea 10 Let Q R d be an integral polytope. Let M R be such that Q {x R d : x i M for i = 1,...,d }. Let x Q and let δ > 0. Then every lattice 1 k Zd for k 2 δ dm contains a lattice point x Q 1 k Zd with x x δ.

8 8 J.A. De Loera, R. Heecke, M. Köppe, R. Weisantel z Z 3 2 (x,z ) 1 δ ( x δ,z ) x R Fig. 2 The principle of grid approxiation. Since we can refine the grid only in the direction of the continuous variables, we need to construct an approxiating grid point (x,z ) in the sae integral slice as the target point (x,z ). Proof By Carathéodory s Theore, there exist d +1 vertices x 0,...,x d Z d of Q and convex ultipliers λ 0,...,λ d such that x = d i=0 λ ix i. Let λ i := 1 k kλ i 0 for i = 1,...,d and λ 0 := 1 d i=1 λ i 0. Moreover, we conclude λ i λ i 1 k for i = 1,...,d and λ 0 λ 0 = d i=1 (λ i λ i ) d 1 k. Then x := d i=0 λ i xi Q 1 k Zd, and we have x x which proves the lea. d i=0 λ i λ i x i 2d 1 k M δ, 4 Bounding techniques Using the results of section 3 we are now able to approxiate the ixed-integer optial point by a point of a suitably fine lattice. The question arises how we can use the geoetric distance of these two points to estiate the difference in objective function values. We prove Lea 11 that provides us with a local Lipschitz constant for the polynoial to be axiized. Lea 11 (Local Lipschitz constant) Let f be a polynoial in d variables with axiu total degree D. Let C denote the largest absolute value of a coefficient of f. Then there exists a Lipschitz constant L such that f (x) f (y) L x y for all x i, y i M. The constant L is O(D d+1 CM D ). Proof Let f (x) = α D c α x α, where D Z d 0 is the set of exponent vectors of onoials appearing in f. Let r = D be the nuber of onoials of f. Then we have f (x) f (y) c α x α y α. α 0

9 FPTAS for ixed-integer polynoial optiization in fixed diension 9 We estiate all suands separately. Let α 0 be an exponent vector with n := d i=1 α i D. Let α = α 0 α 1 α n = 0 be a decreasing chain of exponent vectors with α i 1 α i = e j i for i = 1,...,n. Let β i := α α i for i = 0,...,n. Then x α y α can be expressed as the telescope su x α y α = x α0 y β 0 x α1 y β 1 + x α1 y β 1 x α2 y β 2 + x αn y β n n = (x αi 1 y β i 1 x αi y β i) = i=1 n i=1 ( (x ji y ji )x αi y β i 1). Since x α i y β i 1 M n 1 and n D, we obtain thus x α y α D x y M n 1, f (x) f (y) CrDM D 1 x y. Let L := CrDM D 1. Now, since r = O(D d ), we have L = O(D d+1 CM D ). Moreover, in order to obtain an FPTAS, we need to put differences of function values in relation to the axiu function value. To do this, we need to deal with the special case of polynoials that are constant on the feasible region; here trivially every feasible solution is optial. For non-constant polynoials, we can prove a lower bound on the axiu function value. The technique is to bound the difference of the iniu and the axiu function value on the ixedinteger set fro below; if the polynoial is non-constant, this iplies, for a nonnegative polynoial, a lower bound on the axiu function value. We will need a siple fact about the roots of ultivariate polynoials. Lea 12 Let f Q[x 1,...,x d ] be a polynoial and let D be the largest power of any variable that appears in f. Then f = 0 if and only if f vanishes on the set {0,...,D} d. Proof This is a siple consequence of the Fundaental Theore of Algebra. See, for instance, [4, Chapter 1, 1, Exercise 6 b]. Lea 13 Let f Q[x 1,...,x d ] be a polynoial with axiu total degree D. Let Q R d be an integral polytope of diension d d. Let k Dd. Then f is constant on Q if and only if f is constant on Q 1 k Zd.

10 10 J.A. De Loera, R. Heecke, M. Köppe, R. Weisantel x 0 x 2 x 1 Fig. 3 The geoetry of Lea 13. For a polynoial with axiu total degree of 2, we construct a refineent 1 k Zd (sall circles) of the standard lattice (large circles) such that P 1 k Zd contains an affine iage of the set {0,1,2} d (large dots). Proof Let x 0 Q Z d be an arbitrary vertex of Q. There exist vertices x 1,...,x d Q Z d such that the vectors x 1 x 0,...,x d x 0 Z d are linearly independent. By convexity, Q contains the parallelepiped { S := x 0 + d i=1 λ i(x i x 0 ) : λ i [0, 1 d ] for i = 1,...,d }. We consider the set { S k = 1 k Zd S x 0 + d i=1 n i k (xi x 0 ) : n i {0,1,...,D} for i = 1,...,d }; see Figure 3. Now if there exists a c R with f (x) = c for all x Q 1 k Zd, then all the points in S k are roots of the polynoial f c, which has only axiu total degree D. By Lea 12 (after an affine transforation), f c is zero on the affine hull of S k ; hence f is constant on the polytope Q. Theore 14 Let f Z[x 1,...,x d1,z 1,...,z d2 ]. Let P be a rational convex polytope, and let be the nuber fro Lea 9. Let = k with k Dd 1, k Z. Then f is constant on the feasible region P ( R d 1 Z d 2 ) if and only if f is constant on P ( 1 Z d 1 Z d 2 ). If f is not constant, then f (xax,z ax ) f (x in,z in ) D, (11) where (x ax,z ax ) is an optial solution to the axiization proble over the feasible region P ( R d 1 Z d 2 ) and (x in,z in ) is an optial solution to the iniization proble. Proof Let f be constant on P ( 1 Z d 1 Z d 2 ). For fixed integer part z Z d 2, we consider the polytope P z = { x : (x,z) P }, which is a slice of P scaled to becoe an integral polytope. By applying Lea 13 with k = (D + 1)d on every polytope P z, we obtain that f is constant on every slice P z. Because f is also

11 FPTAS for ixed-integer polynoial optiization in fixed diension 11 Optial ixed-integer solution f (x,z ) Optial grid solution f (x,z ) Rounded ixed-integer solution f ( x δ,z ) Lδ ε 2 f (x,z ) ε 2 f (x,z ) ε 2 f (x,z ) Approxiative grid solution f (x ε/2,z ε/2 ) Fig. 4 Estiates in the proof of Theore 1 (a) constant on the set P ( 1 Z d 1 Z d 2 ), which contains a point of every non-epty slice P z, it follows that f is constant on P. If f is not constant, there exist (x 1,z 1 ), (x 2,z 2 ) P ( 1 Z d 1 Z d 2 ) with f (x 1,z 1 ) f (x 2,z 2 ). By the integrality of all coefficients of f, we obtain the estiate f (x 1,z 1 ) f (x 2,z 2 ) D. Because (x 1,z 1 ), (x 2,z 2 ) are both feasible solutions to the axiization proble and the iniization proble, this iplies (11). 5 Proof of Theore 1 Now we are in the position to prove the ain result. Proof (Proof of Theore 1) Part (a). Let (x,z ) denote an optial solution to the ixed-integer proble (1). Let ε > 0. We show that, in tie polynoial in the input length, the axiu total degree, and ε 1, we can copute a point (x,z) that satisfies (1b 1d) such that f (x,z) f (x,z ) ε f (x,z ). (12) We prove this by establishing several estiates, which are illustrated in Figure 4. First we note that we can restrict ourselves to the case of polynoials with integer coefficients, siply by ultiplying f with the least coon ultiple of all denoinators of the coefficients. We next establish a lower bound on f (x,z ). To this end, let be the integer fro Lea 9, which can be coputed in polynoial tie. By Theore 14 with = Dd 1, either f is constant on the feasible region, or f (x,z ) (Dd 1 ) D, (13) where D is the axiu total degree of f. Now let δ := ε 2(Dd 1 ) D L(C,D,M) (14)

12 12 J.A. De Loera, R. Heecke, M. Köppe, R. Weisantel and let us choose the grid size 4 := ε (Dd 1 ) D L(C,D,M)d 1 M, (15) where L(C,D,M) is the Lipschitz constant fro Lea 11. Then we have 2 δ d 1M, so by Theore 8, there is a point ( x δ,z ) P ( 1 Z d 1 Z d ) 2 with x δ x δ. Let (x,z ) denote an optial solution to the grid proble (7). Because ( x δ,z ) is a feasible solution to the grid proble (7), we have f ( x δ,z ) f (x,z ) f (x,z ). (16) Now we can estiate f (x,z ) f (x,z ) f (x,z ) f ( x δ,z ) L(C,D,M) x x δ L(C,D,M)δ = ε 2 (Dd 1 ) D ε 2 f (x,z ), (17) where the last estiate is given by (13) in the case that f is not constant on the feasible region. On the other hand, if f is constant, the estiate (17) holds trivially. By Corollary 6 we can copute a point (x ε/2,z ε/2 ) P ( 1 Z d 1 Z d 2 ) such that (1 ε 2 ) f (x,z ) f (x ε/2,z ε/2 ) f (x,z ) (18) in tie polynoial in log, the encoding length of f and P, the axiu total degree D, and 1/ε. Here log is bounded by a polynoial in logm, D and logc, so we can copute (x ε/2,z ε/2 ) in tie polynoial in the input size, the axiu total degree D, and 1/ε. Now, using (18) and (17), we can estiate f (x,z ) f (x ε/2,z ε/2 ) f (x,z ) (1 ε 2 ) f (x,z ) = ε 2 f (x,z ) + (1 ε 2 )( f (x,z ) f (x,z ) ) ε 2 f (x,z ) + ε 2 f (x,z ) = ε f (x,z ). Hence f (x ε/2,z ε/2 ) (1 ε) f (x,z ). Part (b). Let the diension d 2 be fixed. We prove that there does not exist a PTAS for the axiization of arbitrary polynoials over ixed-integer sets of polytopes. We use the NP-coplete proble AN1 on page 249 of [8]. This is to decide whether, given three positive integers a, b, c, there exists a positive integer x < c such that x 2 a (od b). This proble is equivalent to asking whether the

13 FPTAS for ixed-integer polynoial optiization in fixed diension 13 axiu of the quartic polynoial function f (x,y) = (x 2 a by) 2 over the lattice points of the rectangle { P = (x,y) : 1 x c 1, 1 a y (c } 1)2 a b b is zero or not. If there existed a PTAS for the axiization of arbitrary polynoials over ixed-integer sets of polytopes, we could, for any fixed 0 < ε < 1, copute in polynoial tie a solution (x ε,y ε ) P Z 2 with f (x ε,y ε ) f (x,y ) ε f (x,y ), where (x,y ) denotes an optial solution. Thus, we have f (x ε,y ε ) = 0 if and only if f (x,y ) = 0; this eans we could solve the proble AN1 in polynoial tie. 6 Extension to arbitrary polynoials In this section we drop the requireent of the polynoial being positive over the feasible region. As we showed in Theore 1, there does not exist a PTAS for the axiization of an arbitrary polynoial over polytopes in fixed diension. We will instead show an approxiation result like the one in [11], i.e., we copute a solution (x ε,z ε ) such that f (xε,z ε ) f (x ax,z ax ) ε f (xax,z ax ) f (x in,z in ), (19) where (x ax,z ax ) is an optial solution to the axiization proble over the feasible region and (x in,z in ) is an optial solution to the iniization proble. Our algorith has a running tie that is polynoial in the input size, the axiu total degree of f, and ε 1. This eans that while the result of [11] was a weak version of a PTAS (for fixed degree), our result is a weak version of an FPTAS (for fixed diension). The approxiation algoriths for the integer case (Theore 5) and the ixedinteger case (Theore 1) only work for polynoial objective functions that are non-negative on the feasible region. In order to apply the to an arbitrary polynoial objective function f, we need to add a constant ter to f that is large enough. As proposed in [6], we can use linear prograing techniques to obtain a bound M on the variables and then estiate f (x) rcm D =: L 0, where C is the largest absolute value of a coefficient, r is the nuber of onoials of f, and D is the axiu total degree. However, the range f (xax,z ax ) f (x in,z in ) can be exponentially sall copared to L0, so in order to obtain an approxiation (x ε,z ε ) satisfying (19), we would need an (1 ε )-approxiation to the proble of axiizing g(x,z) := f (x,z) L 0 with an exponentially sall value of ε. To address this difficulty, we will first apply an algorith which will copute an approxiation [L i,u i ] of the range [ f (x in,z in ), f (x ax,z ax )] with constant quality. To this end, we first prove a siple corollary of Theore 1.

14 14 J.A. De Loera, R. Heecke, M. Köppe, R. Weisantel Corollary 15 (Coputation of upper bounds for ixed-integer probles) Let the diension d = d 1 + d 2 be fixed. Let P R d be a rational convex polytope. Let f Z[x 1,...,x d1,z 1,...,z d2 ] be a polynoial function with integer coefficientsand axiu total degree D that is non-negative on P ( R d 1 Z d 2 ). Let δ > 0. There exists an algorith with running tie polynoial in the input size, D, and 1 δ for coputing an upper bound u such that f (x ax,z ax ) u (1 + δ) f (x ax,z ax ), (20) where (x ax,z ax ) is an optial solution to the axiization proble of f over P ( R d 1 Z d 2 ). Proof Let ε = δ 1+δ. By Theore 1, we can, in tie polynoial in the input size, D, and ε 1 = δ, copute a solution (x ε,z ε ) with f (xax,z ax ) f (x ε,z ε ) ε f (xax,z ax ). (21) Let u := 1 1 ε f (x ε,z ε ) = (1 + δ) f (x ε,z ε ). Then and f (x ax,z ax ) 1 1 ε f (x ε,z ε ) = u (22) (1 + δ) f (x ax,z ax ) (1 + δ) f (x ε,z ε ) = (1 + δ)(1 ε)u ( = (1 + δ) This proves the estiate (20). 1 δ 1 + δ ) u = u. (23) Algorith 16 (Range approxiation) Input: Mixed-integer polynoial optiization proble (1), a nuber 0 < δ < 1. Output: Sequences {L i }, {U i } of lower and upper bounds of f over the feasible region P ( R d 1 Z d 2 ) such that L i f (x in,z in ) f (x ax,z ax ) U i (24) and li U i L i = c ( f (x ax,z ax ) f (x in,z in ) ), (25) i where c depends only on the choice of δ. 1. By solving 2d linear progras over P, we find lower and upper integer bounds for each of the variables x 1,...,x d1,z 1,...,z d2. Let M be the axiu of the absolute values of these 2d nubers. Thus x i, z i M for all i. Let C be the axiu of the absolute values of all coefficients, and r be the nuber of onoials of f (x). Then L 0 := rcm D f (x,z) rcm D =: U 0, as we can bound the absolute value of each onoial of f (x) by CM D.

15 FPTAS for ixed-integer polynoial optiization in fixed diension Let i := Using the algorith of Corollary 15, copute an upper bound u for the proble ax g(x,z) := f (x,z) L i s.t. (x,z) P ( R d 1 Z d ) 2 that gives a (1 + δ)-approxiation to the optial value. Let U i+1 := L i + u. 4. Likewise, copute an upper bound u for the proble ax h(x,z) := U i f (x,z) s.t. (x,z) P ( R d 1 Z d ) 2 that gives a (1 + δ)-approxiation to the optial value. Let L i+1 := U i u. 5. i := i Go to 3. Lea 17 Algorith 16 is correct. For fixed 0 < δ < 1, it coputes the bounds L n, U n satisfying (24) and (25) in tie polynoial in the input size and n. Proof We have and This iplies U i L i+1 (1 + δ) ( U i f (x in,z in ) ) (26) U i+1 L i (1 + δ) ( f (x ax,z ax ) L i ). (27) Therefore U i+1 L i+1 δ(u i L i ) + (1 + δ) ( f (x ax,z ax ) f (x in,z in ) ). ( n 2 ) ( U n L n δ n (U 0 L 0 ) + (1 + δ) δ i f (xax,z ax ) f (x in,z in ) ) i=0 = δ n (U 0 L 0 ) + (1 + δ) 1 δ n 1 ( f (xax,z ax ) f (x in,z in ) ) 1 + δ 1 δ 1 δ ( f (xax,z ax ) f (x in,z in ) ) (n ). The bound on the running tie requires a careful analysis. Because in each step the result u (a rational nuber) of the bounding procedure (Corollary 15) becoes part of the input in the next iteration, the encoding length of the input could grow exponentially after only polynoially any steps. However, we will show that the encoding length only grows very slowly. First we need to reark that the auxiliary objective functions g and h have integer coefficients except for the constant ter, which ay be rational. It turns out that the estiates in the proof of Theore 1 (in particular, the local Lipschitz constant L and the lower bound on the optial value) are independent fro the constant ter of the objective function. Therefore, the sae approxiating grid 1 Zd 1 Z d 2 can be chosen in all iterations of Algorith 16; the nuber only

16 16 J.A. De Loera, R. Heecke, M. Köppe, R. Weisantel depends on δ, the polytope P, the axiu total degree D, and the coefficients of f with the exception of the constant ter. The construction in the proof of Corollary 15 obtains the upper bound u by ultiplying the approxiation f (x ε,z ε ) by (1 + δ). Therefore we have U i+1 = L i + u = L i + (1 + δ) ( f (x ε,z ε ) L i ) = δl i + (1 + δ) f (x ε,z ε ). (28) Because the solution (x ε,z ε ) lies in the grid 1 Zd 1 Z d 2, the value f (x ε,z ε ) is an integer ultiple of D. This iplies that, because L 0 f (x ε,z ε ) U 0, the encoding length of the rational nuber f (x ε,z ε ) is bounded by a polynoial in the input size of f and P. Therefore the encoding length U i+1 (and likewise L i+1 ) only increases by an additive ter that is bounded by a polynoial in the input size of f and P. We are now in the position to prove Theore 2. Proof (Proof of Theore 2) Clearly we can restrict ourselves to polynoials with integer coefficients. Let = (D+1)d 1, where is the nuber fro Theore 8. We apply Algorith 16 using 0 < δ < 1 arbitrary to copute bounds U n and L n for n = log δ ( 2 D (U 0 L 0 ) ). Because n is bounded by a polynoial in the input size and the axiu total degree D, this can be done in polynoial tie. Now, by the proof of Lea 17, we have U n L n δ n (U 0 L 0 ) + (1 + δ) 1 δ n 1 ( f (xax,z ax ) f (x in,z in ) ) 1 δ 1 2 D δ ( f (xax,z ax ) f (x in,z in ) ). (29) 1 δ If f is constant on P ( R d 1 Z d ) 2, it is constant on P ( 1 Z d 1 Z d ) 2, then U n L n 1 2 D. Otherwise, by Theore 14, we have U n L n f (x ax,z ax ) f (x in,z in ) D. This settles part (a). For part (b), if f is constant on P ( R d 1 Z d ) 2, we return an arbitrary solution as an optial solution. Otherwise, we can estiate further: ( 1 U n L n δ ) ( f (xax,z ax) f (x ) in,z in). (30) 1 δ Now we apply the algorith of Theore 1 to the axiization proble of the polynoial function f := f L n, which is non-negative over the feasible region P ( R d 1 Z d 2 ). We copute a point (x ε,z ε ) where ε = ε ( δ 1 δ ) 1 such that f (x ε,z ε ) f (x ax,z ax ) ε f (x ax,z ax ).

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