INTEGRAL DECOMPOSITION OF POLYHEDRA AND SOME ...

INTEGRAL DECOMPOSITION OF POLYHEDRA AND SOME APPLICATIONS IN MIXED INTEGER PROGRAMMING ¨ MARTIN HENK, MATTHIAS KOPPE, AND ROBERT WEISMANTEL Abstract. This paper addresses the question of decomposing an infinite family of rational polyhedra in an integer fashion. It is shown that there is a finite subset of this family that generates the entire family. Moreover, an integer analogue of Carathéodory’s theorem carries over to this general setting. The integer decomposition of a family of polyhedra has different applications in integer and mixed integer programming. Three of them will be illuminated.

1. Introduction This paper deals with the question of decomposing an infinite family of rational polyhedra that one associates with a fixed matrix and varying right hand side vectors into irreducible ones. This problem has been studied in various fields of geometry when the right hand side vectors are arbitrary vectors. A major technique that is used in this context is the Minkowski sum of sets, i.e., the pointwise addition of points from different sets. Our object of investigation is in fact the integral decomposition of polyhedra. More precisely, given a fixed matrix and an infinite family of integral right hand side vectors, we are interested in a subset of this family of integral right hand sides that generate the entire family of polyhedra with respect to an operation that is to be specified according to the concrete application. It is proved that there always exists such a generating subset of finite cardinality. This extends naturally the result about the existence of a Hilbert basis for rational polyhedral cones. Here the family of polyhedra that one considers is just a family of points, i.e., a family of 0-dimensional polyhedra. Recall that the Hilbert basis of a rational polyhedral cone C is a minimal finite generating set of the integral points in C with respect to non-negative integral combinations. In other words, the Hilbert basis H(C) consists of a finite number of vectors h1 , . . . , hl ∈ C ∩ Zd \ {0} such that n o C ∩ Zd = λ1 h1 + λ2 h2 + · · · + λl hl : λi ∈ Z+ , 1 ≤ i ≤ l . Date: February 4, 2007. 1991 Mathematics Subject Classification. 52C17, 11H31. Key words and phrases. mixed integer programming, test sets, indecomposable polyhedra, Hilbert bases, rational polyhedral cones. This work was supported partially by the DFG through grant WE1462, by the Kultusministerium of Sachsen Anhalt through the grants FKZ37KD0099 and FKZ 2945A/0028G and by the EU Donet project ERB FMRX-CT98-0202. 1

2

¨ MARTIN HENK, MATTHIAS KOPPE, AND ROBERT WEISMANTEL

The existence of a Hilbert basis follows from the classical lemma of Gordan [Go1873], and it was shown by van der Corput [Cor31] that the Hilbert basis of a pointed cone C is uniquely determined by the following characterization. Lemma 1.1 (van der Corput). The Hilbert basis of a rational polyhedral pointed cone C ⊂ Rn consists of all elements h ∈ C ∩ Zn which can not be written as h = z 1 + z 2 for some z i ∈ C ∩ Zn \ {0}. For further geometric and algorithmic properties of Hilbert bases we refer to [AWW00]. In the same vein it is shown in this paper that an integer analogue of Carathéodory’s theorem that is known to hold for the integral points in a rational polyhedral cone carries over to the more general setting when one deals with families of polyhedra of arbitrary dimension with integral right hand sides. The question of decomposing such a family of polyhedra with integral right hand sides in an integer fashion has different applications in integer and mixed integer programming. In the sequel we will illuminate three such settings that we consider relevant. Application 1 addresses the question of whether there exists a finite certificate for testing that an infinite family of polyhedra with integral right hand sides is integral. Note that a rational polyhedron is called integral if each of its faces contains an integral point. To be more precise, let A ∈ Zm×n denote a given matrix and b ∈ Zm a vector. By Pb = { x ∈ Rn : Ax ≤ b } we denote the polyhedron associated with A and b. Let L denote a lattice in Zm . Theorem 1.1. There exists a finite subset F ⊆ L such that Pb is integral for all b ∈ L if and only if Pb is integral for all b ∈ F. Theorem 1.1 follows immediately from the theory of integral decomposition of polyhedra that we will introduce in Chapter 3. Application 2 extends the first application by considering totally dual integral systems of inequalities instead of integer polyhedra. Definition 1.1. Let A ∈ Zm×n , b ∈ Zm . The system of inequalities Ax ≤ b is called totally dual integral (TDI) if for every c ∈ Zm such that |min{ b⊺y : A⊺y = c, y ≥ 0 }| < ∞, there exists an integral vector y ∗ ∈ Zm , A⊺y ∗ = c, y ∗ ≥ 0 such that b⊺y ∗ = min{ b⊺y : A⊺y = c, y ≥ 0 }. Theorem 1.2. There exists a finite subset F ⊆ L such that Ax ≤ b is TDI for all b ∈ L if and only if Ax ≤ b is TDI for all b ∈ F. In this sense, Theorem 1.2 shows the existence of a a finite certificate for testing that an infinite family of systems of inequalities with integral right hand sides is TDI. This is indeed a generalization of Theorem 1.1 because if Ax ≤ b is TDI and b is integral, then { x ∈ Rn : Ax ≤ b } is integral, however, the converse is not true, see [EG77].

INTEGRAL DECOMPOSITION OF POLYHEDRA

3

Application 3 of the method of decomposing integral polyhedra in an integer fashion is directed towards the design of a primal approach for mixed integer programming. For integral matrices A ∈ Zm×d , B ∈ Zm×n and integral vectors b ∈ Zm , α ∈ Zd and β ∈ Zn the mixed integer linear programming problem (MIP) is the task to determine max α⊺x + β ⊺y : A x + B y ≤ b, x ∈ Zd , y ∈ Rn . (1.1) By a primal approach we mean an augmentation algorithm that starting from any feasible point of the mixed integer program moves through an improving direction to a new feasible point of the program as long as possible. Of course, the set of all improving directions for all the feasible points of even a single pure integer program may not be finite. However, in the pure integer case, one can resort to a finite generating set for the family of all improving directions for an integer program. Such a set is commonly called a test set. A particularly nice geometric way of defining a test set has been presented by Graver [Gra75]. Definition 1.2. For the family of integer programs of the form (1.1) with n = 0 associated with a fixed matrix A but varying data b ∈ Zm , α ∈ Rd , the Graver test set G(A) is defined as n o [ G(A) = H x ∈ Rd : Aε x ≤ 0 , (1.2) ε∈{−1,1}m

where Aε arises from A by multiplying the i-th row A with εi . One purpose of this note is to present an analogous test set approach for mixed integer linear programming problems. Here, the situation is however significantly more complicated, because a minimal test set with respect to inclusion may not be finite as the following trivial example demonstrates.

x 1

1 y Figure 1. A mixed integer linear program with an infinite minimal test set Example 1. We consider the mixed integer program max s. t.

x x + y ≤ 1 −x ≤ 0 −y ≤ 0 x ∈ Z, y ∈ R

(1.3)

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whose feasible set is the points of the unit triangle with integral x coordinate, i.e., a single point (1, 0) with objective value 1 and the points (0, y) for y ∈ [0, 1] with objective value 0; see Figure 1. Every test set for (1.3) must contain all the points (1, −y) for y ∈ [0, 1]. The way to overcome this difficulty is to resort to a finite representation of the infinite test set. This can indeed be done by applying the methods that we introduce in the context of integral decompositions of rational polyhedra. The proof of the following result will then follow. Theorem 1.3. There exists a finite set G(A, B) ⊂ { (x, ε, u)⊺ : x ∈ Zd , ε ∈ {−1, 1}m , u ∈ Zm }, depending only on A and B, such that for each non optimal point (x, y)⊺ ∈ {(x, y)⊺ ∈ Zd × Rn : A x + B y ≤ b} of (1.1) there exists a (e x, ε, u)⊺ ∈ G(A, B) and a vector ye ∈ Puε satisfying e) + B (y + ye) ≤ b, i) A (x + x

where Puε = { y ∈ Rn : Bε y ≤ u }.

ii) α⊺(x + x e) + β ⊺(y + ye) > α⊺x + β ⊺y,

2. Linear decomposition of polytopes In the following let W ∈ Zm×n be a fixed but arbitrary integral matrix with row vectors wi ∈ Zn , 1 ≤ i ≤ m. We assume that pos{w1 , . . . , wm } = Rn , where pos denotes the positive hull. Thus for every u ∈ Rm the set Pu = { y ∈ Rn : W y ≤ u } is a polytope. Now we are interested in the family of all non empty polytopes arising in this way and therefore we set U(W ) = { u ∈ Rm : Pu 6= ∅ } . This set has been investigated by various authors in different contexts (cf. [Gr¨ u67, p. 316], [KLS90], [McM73], [Mey74], [Smi87] and the references within). It is not hard to see that the dual set U ∗ (W ) = { s ∈ Rm : s⊺u ≥ 0, for all u ∈ U(W ) } is given by U ∗ (W ) = { s ∈ Rm : W ⊺s = 0 and s ≥ 0 }. This shows, in particular, that U(W ) is a rational polyhedral m-dimensional cone. Since we do not want to distinguish between a polytope Pu and its translate t + Pu = Pu+W t for t ∈ Rn , we choose as a representative of each equivalence class Pu+W t , t ∈ Rn , the right hand side u ˜ satisfying W ⊺u ˜ = 0. In other words, u ˜ is the orthogonal projection of u onto the orthogonal complement of the space W Rn . Hence e ) = u ∈ Rm : Pu 6= ∅ and W ⊺u = 0 U(W is a rational polyhedral (m − n)-dimensional cone and we have e ) ⊕ W Rn . U(W ) = U(W

Since the maximal linear subspace of U(W ) consists of the space W Rn e ) is even pointed. Moreover, the dual cone Ue∗ (W ) of U(W e ) the cone U(W m ⊺ w.r.t. the subspace { s ∈ R : W s = 0 } reads e ), W ⊺s = 0 . Ue∗ (W ) = s ∈ Rm : s⊺u ≥ 0 for all u ∈ U(W


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This shows that Ue∗ (W ) coincides with U ∗ (W ). By the primal-dual relatione ). ship we obtain the following characterization of the facets of U(W

Remark 2.1. For s ∈ Ue∗ (W ) let Ws be the matrix consisting of all rows wi with si > 0 and let rs be the number of rows. Then the outer normal vector e ) are given by { s ∈ Ue∗ (W ) : rank(Ws ) = rs − 1 }. of the facets of U(W

Let us mention that there is also a nice geometrical meaning of the outer e ). Namely, via the so called Minkowski’s normal vectors of the facets of U(W existence theorem of polytopes [Mi1897] one can assign to each outer normal vector s a (#Ws − 1)-dimensional unique (up to dilations) simplex such that e ) can be written as the Blaschke sum of at each polytope Pu for u ∈ U(W most m − n of those simplices (see [Gr¨ u67, p. 331]). e ) arises from the study of (in)decomAnother interesting aspect of U(W posable polytopes. Two polytopes P1 , P2 ⊂ Rn are called homothetic if P1 = ρP2 + t for some t ∈ Rn and ρ > 0. A polytope P ⊂ Rn is called decomposable if two polytopes P1 and P2 exist with P = P1 + P2 , and Pi , i = 1, 2, are not homothetic to P . Otherwise P is indecomposable. Here P1 + P2 denotes the usual Minkowski sum of two sets. Observe that, in this general setting, every point is indecomposable. Moreover a polytope P1 is called a summand of a polytope P (P1 ≺ P ) if there exist a positive scalar ρ and a polytope P2 such that P = ρP1 + P2 . For a polytope Pu , u ∈ U(W ), each summand admits a representation as Pv for a certain v ∈ U(W ). Again, since we may neglect translations, e ), is decomposable if and only if there exist the polytope Pu , u ∈ U(W e ) such that Pu = Pu1 +Pu2 and Pui , i = 1, 2, are not homothetic u1 , u2 ∈ U(W e ). to Pu . In fact, one can even establish a stronger relation to the cone U(W To this end we denote for u ∈ U(W ) by η(u) ∈ U(W ) the support vector of the polytope Pu , i.e., η(u)i = max (wi )⊺y : y ∈ Pu , 1 ≤ i ≤ m. η(u) is the componentwise least right hand side which yields the same polytope, Pu = Pη(u) . One can show that (cf. e.g. [Gr¨ u67]) Pu = Pu1 + Pu2 ⇔ η(u) = η(u1 ) + η(u2 ) and Pu1 , Pu2 ≺ Pu .

(2.1)

Furthermore, it follows from works of McMullen [McM73] and Meyer [Mey74] e )u = { η(v) : v ∈ U(W e ) with Pv ≺ Pu } is a rational polyhedral that U(W e ), where the extreme rays of U(W e )u correspond to indesubcone of U(W composable polytopes. As a nice consequence of this construction we get by Carathéodory’s theorem and (2.1) e ) can be Theorem 2.1 (McMullen, Meyer). Every polytope Pu for u ∈ U(W written as the Minkowski sum of at most m − n indecomposable polytopes. Indeed, depending on certain affine dependencies of the rows of the matrix W one can give stronger bounds (cf. [Mey74], [McM73], [Smi87]). The e )u consists of all polytopes Pv which relative interior of this cone U(W are strongly combinatorially equivalent to Pu . Recall that two polytopes

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P, Q ⊂ Rn are called strongly combinatorially equivalent if for all v ∈ Rn the following equality holds: dim y ∈ P : v ⊺y = max{ v ⊺y : y ∈ P } = dim y ∈ Q : v ⊺y = max{ v ⊺y : y ∈ Q } .

e ) possesses finitely many strongly combinatorially non Since the cone U(W equivalent polytopes we can extend Theorem 2.1 to the cone U(W ). Corollary 2.1. There exist finitely many vectors u1 , . . . , uk ∈ U(W ), corresponding to indecomposable polytopes, such that each polytope Pu for u ∈ U(W ) can be written as a non-negative linear combination of at most m of the polytopes Pui . Proof. By the foregoing remarks and Theorem 2.1 we know that there exe ), corresponding to indecomposable polyist finitely many u1 , . . . , ul ∈ U(W e ) the polytope Pu˜ can be written as a nontopes, such that for each u ˜ ∈ U(W negative linear combination of at most (m − n) polytopes Pui , i ∈ {1, . . . , l}. Let us select ul+1 , . . . , ul+n+1 ∈ W Rn such that pos{ul+1 , . . . , ul+n+1 } = W Rn . Obviously, Pul+i is indecomposable. Let u ∈ U(W ) and let u ˜ ∈ e U(W ) be its orthogonal projection onto the orthogonal complement of W Rn . Then there exist non-negative scalars λi such that u − u ˜ = λ1 ul+1 + · · · + l+n+1 λn+1 u , where at least one λi vanishes. We have Pu = Pu˜ + λ1 Pul+1 + · · · λn+1 Pul+n+1 and hence the vectors u1 , . . . , ul+n+1 have the desired property. So one may say that the vectors ui from the corollary above form a minimal generating system of U(W ) (or of all polytopes Pu for u ∈ U(W )). 3. Integral decomposition of polytopes Here we want to study an analogous relation for polytopes Pu with an integral right hand side, i.e., u ∈ U(W ) ∩ Zm , and with respect to nonnegative integral combinations. To this end we need one more result on Hilbert bases of rational polyhedral pointed cones that we will refer to as the weak integer Carathéodory property. Theorem 3.1 (Seb¨ o, [Seb90]). Each integral vector of a rational polyhedral pointed cone C ⊂ Rn can be written as a non-negative integral combination of at most 2n − 2 elements of its Hilbert basis H(C). Moreover, it was also shown by Seb¨ o that in the 3-dimensional case 3 elements of the Hilbert basis are sufficient. However, there does not exist a strong integral counterpart to Carathéodory’s theorem since, in general, one needs more elements of the Hilbert basis than the dimension of the cone [B-W99]. In order to establish an analogous statement to Corollary 2.1 we need to express precisely what we mean by the integral decomposition of a polytope Pz , z ∈ U(W ) ∩ Zm . In view of our application to test sets we define Definition 3.1. A polytope Pz , z ∈ U(W ) ∩ Zm , is called integral decomposable if there exist Pz 1 , Pz 2 not homothetic to Pz such that Pz = Pz 1 + Pz 2


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and z = z 1 + z 2 , z i ∈ U(W ) ∩ Zm . Otherwise Pz is called integral indecomposable. Observe that in contrast to the usual definition of indecomposability this definition does not depend on the polytope only but also on the right hand side, i.e., on the representation of the polytope. It may indeed happen that for some z 1 , z 2 ∈ U(W ) ∩ Zm the polytopes Pz 1 and Pz 2 coincide, although Pz 1 is decomposable, whereas Pz 2 is indecomposable. Of course, the reason lies in the representation of redundant facets. These do not play any role in the non integral case, since we may always fix the representation of the polytope via the support vector. If we restrict our attention to polytopes Pz , z ∈ U(W ) ∩ Zm , with integral support vectors, then we have the following Corollary 3.1. There exist finitely many vectors hi ∈ U(W ) ∩ Zm , corresponding to integral indecomposable polytopes, such that for each polytope Pz , z ∈ U(W )∩Zm , with η(z) = z there exist hj1 , . . . , hj2m−2−n and non-negative integers λj1 , . . . , λj2m−2−n with Pz =

2m−2−n X i=1

λji Phji

and

z=

2m−2−n X

λji hji .

i=1

Before giving the proof we fix some notation to which we shall also refer later. Let wc1 , . . . , wcn be the column vectors of the matrix W . Choose points wcn+1 , . . . , wcm ∈ Zm such that the vectors wc1 , . . . , wcm form a basis of the P m i m . Then for each u ∈ U, u = lattice ZP i=1 λi wc , let u be the vector given m i by u = i=n+1 λi wc . We define U(W ) = u : u ∈ U(W ) . (3.1) U(W ) is a rational polyhedral pointed (m − n)-dimensional cone. In particular, we have U(W ) ∩ Zm + W Zn = U(W ) ∩ Zm . We are now ready for the proof of Corollary 3.1.

Proof of Corollary 3.1. Let z ∈ U(W ) ∩ Zm with η(z) = z and thus z = η(z) ∈ Zm . As in the general case we consider the cone U(W )z = η(v) : v ∈ U(W ) and Pv ≺ Pz .

U(W )z is a rational polyhedral pointed cone. Therefore, it possesses a unique Hilbert basis that we denote by h1 , . . . , hk . First we claim, that the polytopes Phi are integral indecomposable. For if not, then there exist z 1 , z 2 ∈ U(W ) ∩ Zm \ {0} such that Phi = Pz 1 + Pz 2 and hi = z 1 + z 2 . This yields (see (2.1)) z 1 + z 2 = hi = η(hi ) = η(z 1 ) + η(z 2 ) ≤ z 1 + z 2 . In other words, η(z 1 ) = z 1 and η(z 2 ) = z 2 . Thus z 1 , z 2 belong to the cone U(W )z . This contradicts the definition of a Hilbert basis, cf. Theorem 1.1.

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On account of (2.1) and Theorem (3.1), there exist hji and non-negative integers λji , 1 ≤ i ≤ 2(m − n) − 2, such that Pz =

2m−2−n X

λji Phji

and z =

2m−2−n X

λji hji .

i=1

i=1

Finally, we note that z − z can be written as a non-negative integral combination of at most n vectors from the set wc1 , . . . , wcn , −(wc1 + · · · + wcn ). As mentioned before, the relative interior points of the cone U(W )z correspond to polytopes which are strongly combinatorially equivalent to Pz and since there are only finitely many strongly combinatorially non equivalent polytopes Pu , u ∈ U(W ), we are done. In the general case we are not aware of a nice geometric description of a family of indecomposable polytopes such that each polytope can be written as a non-negative integral combination of a fixed number of polytopes. However, a finite generating system for all Pz , z ∈ U(W ) ∩ Zm , with the weak integer Carathéodory property can always be constructed. Theorem 3.2. There exist finitely many vectors g 1 , . . . , g k ∈ U(W ) ∩ Zm such that for each polytope Pz , z ∈ U(W ) ∩ Zm , there exist g j1 , . . . , g j2m−2−n and non-negative integers λj1 , . . . , λj2m−2−n such that Pz =

2m−2−n X

λji Pgji

i=1

and

z=

2m−2−n X

λ ji g ji .

i=1

Proof. For an index set I = {i1 , . . . , in } corresponding to linearly independent row vectors wi1 , . . . , win of the matrix W , let WI be the regular submatrix consisting of these rows and let ΠI be the matrix representing the projection of an m-dimensional vector b onto its coordinates bi1 , . . . , bin . Let I1 , . . . , Ik be all possible index sets of this type and let ⊺ M = W (WI1 )−1 ΠI1 − Em , . . . , W (WIk )−1 ΠIk − Em ∈ Rk m×m ,

where Em is the m × m identity matrix. Let U(W ) be the cone constructed in (3.1) and for a given orthant O in Rk m let U(W )O = { z ∈ U(W ) : M z ∈ O }.

U(W )O is a rational polyhedral pointed cone of dimension at most m − n. Next we claim, Pz 1 + Pz 2 = Pz 1 +z 2 ,

for z 1 , z 2 ∈ U(W )O .

(3.2)

To this end let Ii be the set of all index sets I satisfying W (WI )−1 ΠI z i − z i ≤ 0, and for an arbitrary index set let viI = (WI )−1 ΠI z i , i = 1, 2. In particular we have Pz i = conv viI : I ∈ Ii .


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Let I be an index set such that v I = (WI )−1 ΠI (z 1 + z 2 ) is a vertex of the polytope Pz 1 +z 2 . Of course, we have v I = v1I + v2I and since the vectors W (v1I ), W (v2I ) lie in the same orthant O we get I ∈ I1 ∩ I2 . Thus Pz 1 +z 2 = conv{ v1I + v2I : I ∈ I1 ∩ I2 } ⊂ Pz 1 + Pz 2 ⊂ Pz 1 +z 2 , which shows (3.2). Now let g 1 , . . . , g l be the Hilbert basis of the cone U(W )O . Then by Theorem 3.1 and (3.2) we know that each Pz for z ∈ U(W )O can be written as a non negative integral combination of at most 2(m − n) − 2 polytopes of the form Pgi and the corresponding right hand sides sum up. Finally, let g 1 , . . . , g k be the union of all Hilbert bases with respect to all orthants. Then these vectors form a generating system satisfying the requirements of the theorem for all z ∈ U(W ) ∩ Zm . Finally, as in the proof of Corollary 3.1, we may conclude that g 1 , . . . , g k , w1 , . . . , wn , −(w1 + · · · + wn ) have the desired property for all z ∈ U(W ) ∩ Zm . Next we want to consider a slight variant of the above problem. To this end let W ∈ Zm×n as above and let L ⊂ Zm be a d-dimensional lattice. Then we are interested in the set U(L, W ) = l ∈ L : Pl 6= ∅ . (3.3) In other words we restrict the class of all possible right hand sides to a sublattice. It is not hard to see that also in this case we can easily find via the methods described in the proof of Theorem 3.2 a finite generating system. More precisely we have

Remark 3.1. There exists a finite set of lattice points H(L, W ) ⊂ U(L, W ) such that for each l ∈ U(L, W ) there exists at most 2m − 2 − n elements h1 , . . . , hk ∈ H(L, W ) and positive integers λ1 , . . . , λk such that Pl = λ1 Ph1 + · · · + λk Phk

and

l = λ1 h1 + · · · + λk hk .

4. Proofs of the Theorems The proof of Theorem 1.1 is an immediate consequences of the previous section. Proof of Theorem 1.1. Let H(L, W ) as in Remark 3.1. Then we obviously have that Pl is integral for all l ∈ L if and only if Ph is integral for all h ∈ H(L, W ). For the proof of Theorem 1.2 we need a well-known characterization of TDI-systems due to Giles and Pulleyblank [GP79]. Theorem 4.1. Let A ∈ Zm×n and b ∈ Zm . The system Ax ≤ b is TDI if and only if for every minimal face of Pb the set of row vectors that are tight at this face determine a Hilbert basis of the cone that they generate. Proof of Theorem 1.2. Thereom 4.1 implies that for two strongly combinatorially equivalent polytopes Pl1 and Pl2 , li ∈ Zm ∩ L it holds, that Pl1 is TDI if and only if Pl2 is TDI. Since there are only finitely many strongly


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combinatorially non equivalent polytopes of the type Pl , l ∈ L, we have a finite certificate to test whether all polytopes Pl , l ∈ L, are TDI. Now we come to the proof of the main Theorem. We consider the mixed integer linear program as described in (1.1). Let a1 , . . . , am and b1 , . . . , bm be the row vectors of the matrix A and B. For ε ∈ {−1, 1}m let Aε and Bε be the matrices with row vectors ε1 a1 , . . . , εm am and ε1 b1 , . . . , εm bm , respectively. We may assume that rank(A) = d by replacing the columns of A with a basis for the lattice AZd . Proof of Theorem 1.3. For each ε ∈ {−1, 1}m let     A−ε Bε ¯ε = −En  ,  and B −En A¯−ε =  En En

¯ε ) ⊂ Zd be a finite where En is the n × n identity matrix, and let H(A¯−ε , B generating set of all non empty polytopes of the form ¯ε y ≤ A¯−ε (x, l, u)⊺ } PAε¯−ε (x,l,u)⊺ = { y ∈ Rn : B = { y ∈ Rn : Bε y ≤ A−ε x, l ≤ y ≤ u } for arbitrary x ∈ Zd and lower and upper bounds l, u ∈ Zn as defined in Remark 3.1. Note that PAε¯−ε (x,l,u)⊺ ⊂ PAε −ε x for all l, u ∈ Zn , where PAε −ε x is the polyhedron defined in Theorem 1.3. Finally we set [ ¯ε ), u = A−ε x (x, ε, u) : x ∈ H(A¯−ε , B G(A, B) = ε∈{−1,1}m

and claim that this set has the properties required in the theorem. To this end, let (¯ x, y¯)⊺ be a feasible but non-optimal solution of (MIP) and let ∗ ∗ ⊺ (x , y ) be an optimal solution of the problem. Now let I1 , I2 be a partition of the indices {1, . . . , m} such that (ai )⊺x ¯ + (bi )⊺y¯ ≤ (ai )⊺x∗ + (bi )⊺y ∗ , i ⊺

i ⊺ ∗

i ⊺

i ⊺ ∗

(a ) x ¯ + (b ) y¯ ≥ (a ) x + (b ) y ,

i ∈ I1 , i ∈ I2 .

Define ε ∈ {−1, 1}m according to εi = −1 for i ∈ I1 and εi = 1 for i ∈ I2 . Then we have Bε (y ∗ − y¯) ≤ A−ε (x∗ − x ¯) ⊺ ¯ε ) for some l, u ∈ Zn . Hence there and thus (x∗ − x ¯, l, u) ∈ U(A¯−ε , B ⊺ 1 1 1 k k k ¯ε ) and positive integers exist some (g , l , u ) , . . . , (g , l , u ) ∈ H(A¯−ε , B λ1 , . . . , λk such that j ∗ X k k X g x −¯ x ε j λj PAε¯−ε (gj ,lj ,uj )⊺ . λi l and PA¯−ε (x∗ −¯x,l,u)⊺ = = l u

j=1

uj

j=1

Thus there exist y j ∈ PAε −ε gj such that

x∗ −¯ x y ∗ −¯ y

=

k X j=1

λj

gj yj

.


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Now we show that for each j ∈ {1, . . . , k} the vector (¯ x + g j , y¯ + y j ) is feasible. For i ∈ {1, . . . , m} we have i

i

bi ≥ (a , b )

x∗ y∗

i

i

= (a , b )

By definition we have j g (ai , bi ) yj ≥ 0 for i ∈ I1

and

x ¯ y¯

+

k X

λj (ai , bi )

j=1

(ai , bi )

gj yj

≤0

gj yj

.

(4.1)

for i ∈ I2 .

Since all the scalars in (4.1) are positive, every vector (¯ x + g j , y¯ + y j ) for j ∈ {1, . . . , k} is feasible. Finally we observe that α⊺x∗ + β ⊺y ∗ > α⊺x ¯ + β ⊺y¯. Thus there exists at least one index j such that α⊺g j + β ⊺y j > 0. This finally shows that the vector (g j , y j )⊺ is both applicable at (¯ x, y¯)⊺ and improving. References [AWW00] K. Aardal, R. Weismantel, and L. Wolsey, Non-standard approaches to integer programming, CORE Discussion Paper No. 2000/2 (2000). [B-W99] W. Bruns, J. Gubeladze, M. Henk, A. Martin, and R. Weismantel, A counterexample to an integer analogue of Carathéodory’s theorem, J. reine angew. Math. 510 (1999), 179–185. ¨ [Cor31] J. G. van der Corput, Uber Systeme von linear-homogenen Gleichungen und Ungleichungen, Proceedings Koninklijke Akademie van Wetenschappen te Amsterdam 34 (1931), 368–371. [EG77] J. Edmonds and R. Giles, A min-max relation for submodular functions on graphs, Studies in Integer Programming (P. L. Hammer, ed.), Annals of Discrete Mathematics 1, North-Holland, Amsterdam (1977), 185–204. [GP79] R. Giles and W. R. Pulleyblank, Total dual integrality and integer polyhedra, Linear Algebra and Applications 25 (1979), 191–196. ¨ [Go1873] P. Gordan, Uber die Aufl¨ osung linearer Gleichungen mit reellen Coefficienten, Math. Ann. 6 (1873), 23–28. [Gra75] J. E. Graver, On the foundations of linear and integer linear programming I, Math. Program. 8 (1975), 207–226. [Gr¨ u67] B. Gr¨ unbaum, Convex polytopes, Wiley Interscience, 1967. [KLS90] R. Kannan, L. Lov´ asz, and H. E. Scarf, The shapes of polyhedra, Math. Operations Research 15, No. 2 (1993), 364–380. [McM73] P. McMullen, Representations of polytopes and polyhedral sets, Geometriae Dedicata 2 (1973), 83–99. [Mey74] W. Meyer, Indecomposable polytopes, Trans. Amer. Math. Soc. 190 (1974), 77–86. [Mi1897] H. Minkowski, Allgemeine Lehrs¨ atze u ¨ber die konvexen Polyeder, Nachr. Ges. Wiss. G¨ ottingen (1897), 198–219. (Gesammelte Abhandlungen, Berlin 1911). [Seb90] A. Seb¨ o, Hilbert bases, Carathéodory’s Theorem and combinatorial optimization, Proc. of the IPCO conference, Waterloo, Canada (1990), 431–455. [Smi87] Z. Smilansky, Decomposability of polytopes and polyhedra, Geometriae Dedicata 24 (1987), 29–49. E-mail address: {henk,mkoeppe,weismantel}@imo.math.uni-magdeburg.de ¨tsDepartment of Mathematics/IMO, University of Magdeburg, Universita platz 2, D-39106 Magdeburg