Operations Research Letters 39 (2011) 457–460 Contents lists available at SciVerse ScienceDirect Operations Research Letters journal homepage: www.elsevier.com/locate/orl Integer-empty polytopes in the 0/1-cube with maximal Gomory–Chvátal rank Sebastian Pokutta a, Andreas S. Schulz b,∗ a Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany b Massachusetts Institute of Technology, USA article info a b s t r a c t Article history: We provide a complete characterization of all polytopes P ⊆ T0; 1Un with empty integer hulls, whose Received 6 March 2011 Gomory–Chvátal rank is n (and, therefore, maximal). In particular, we show that the first Gomory–Chvátal Accepted 13 September 2011 closure of all these polytopes is identical. Available online 25 September 2011 ' 2011 Elsevier B.V. All rights reserved. Keywords: Integer programming Cutting planes Gomory–Chvátal closure Rank 1. Introduction polynomial upper bounds on the rank of polytopes in the 0=1-cube (cf., [1,5]) crucially depend on this special case. The improvement The Gomory–Chvátal procedure is a well-known technique to from O.n3 log n/ in [1] to O.n2 log n/ in [5] as an upper bound on T Un derive valid inequalities for the integral hull PI of a polyhedron the rank of polytopes in 0; 1 is a direct consequence of a better n P D fx 2 R j Ax ≤ bg. It was introduced by Chvátal [2] upper bound on the rank of certain polytopes in the 0=1-cube and, implicitly, by Gomory [6–8] as a means to establish certain that do not contain integral points. It can actually be shown that n combinatorial properties via cutting-plane proofs. Cutting planes lower bounds on the rank of polytopes P ⊆ T0; 1U with PI D; and Gomory–Chvátal cuts, in particular, belong to today's standard play a crucial role in understanding the rank of any (well-defined) toolbox in integer programming. However, despite significant cutting-plane procedure [10]. Moreover, in many cases, the rank progress in recent years (see, e.g., [1,3,5,9]), the Gomory–Chvátal of a face F ⊆ P with FI D; induces a lower bound on the rank of procedure is still not fully understood from a theoretical stand- P itself. In fact, the construction of the aforementioned families of T Un point, especially in the context of polytopes contained in the polytopes in 0; 1 whose rank is strictly larger than n exploits this 0=1-cube. For example, the question whether the currently best connection. known upper bound of O.n2 log n/ on the Gomory–Chvátal rank, In view of this, a thorough understanding of the Gomory– Chvátal rank of polytopes P ⊆ T0 1Un with P D; might help established in [5], is tight, remains open. In [5], it was also shown ; I to derive better upper and lower bounds for the general case. In that there is a class of polytopes contained in the n-dimensional this paper, we characterize all polytopes P ⊆ T0; 1Un with P D; 0=1-cube whose rank exceeds n. (See [11] for a more explicit con- I and rk.P/ D n. In particular, we show that after applying the struction.) However, no family of polytopes in the 0=1-cube is Gomory–Chvátal procedure once, one always obtains the same known that realizes super-linear rank, and thus there is a large polytope. Furthermore, we show that P ⊆ T0; 1Un with P D; gap between the best known upper bound and the largest realized I has rk.P/ D n if and only if P \ F 6D ; for all one-dimensional faces rank. F of the 0=1-cube T0; 1Un. ⊆ T Un D; We consider the special case of P 0; 1 with PI The paper is organized as follows. In Section 2, we introduce D and Gomory–Chvátal rank rk.P/ n (i.e., maximal rank, as our notation and recall some basic facts about the Gomory–Chvátal ≤ ⊆ T Un D; rk.P/ n holds for all P 0; 1 with PI ; see [1]). procedure. Afterwards, in Section 3, we derive the characterization This case is of particular interest as, so far, all known proofs of n of all polytopes P ⊆ T0; 1U with PI D; and rk.P/ D n. In parti- cular, in Section 3.2, we relate the rank of a polytope P ⊆ T0; 1Un with PI D; to the rank of its faces. We then prove the charac- ∗ Correspondence to: Massachusetts Institute of Technology, E62-569, 100 Main terization for the two-dimensional case in Section 3.3, which is an Street, Cambridge, MA 02142, USA. essential ingredient for the subsequent generalization to arbitrary E-mail address: [email protected] (A.S. Schulz). dimension in Section 3.4. 0167-6377/$ – see front matter ' 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2011.09.004 458 S. Pokutta, A.S. Schulz / Operations Research Letters 39 (2011) 457–460 2. Preliminaries to deduce .Bn/I D;, and the rank with respect to the classical Gomory–Chvátal procedure coincides with the rank if one were n m×n 1 Let P D fx 2 R j Ax ≤ bg be a polytope with A 2 Z and to use 0; 2 -cuts only. Clearly, with Bn as above and F being a m n ∼ b 2 Z . The Gomory–Chvátal closure of P is defined as k-dimensional face of T0; 1U , we have Bn \ F D Bk. As a direct consequence of the proof of [3, Lemma 7.2] one obtains: 0 \ P VD fx V λAx ≤ bλbcg: m n n λ2RC,λA2Z Lemma 3.1. Let P ⊆ T0; 1U be a polytope with Fk ⊆ P for some 0 k < n. Then FkC1 ⊆ P . The result P0 is again a polytope (see [2]), and one can apply the operator iteratively. We let P.iC1/ VD .P.i//0 for i ≥ 0 and P.0/ VD P. Proof. We include a proof for completeness. Let P as above and let .i/ ax < b C 1 with a 2 n and b 2 be valid for P. We have to show The resulting sequence fP gi≥0 becomes stationary after finitely Z Z C many steps [2], and the smallest k such that P.k 1/ D P.k/ is that ap ≤ b for every p 2 FkC1. Let p 2 FkC1 be arbitrary. If ap 2 Z, the Gomory–Chvátal rank of P (in the following often rank of P), we are done. So assume that ap 62 Z. Then there exists i 2 TnU such 1 0 1 0 .rk.P// that ai 6D 0 and pi D . We define the points p ; p by setting p D denoted by rk.P/. In particular, P D PI , where PI VD conv.P \ 2 j n 1 D 6D 0 D 1 D D 1 0 C 1 1 Z / denotes the integral hull of P. pj pj for all j i; pi 0, and pi 1. Hence, p 2 p 2 p . 0 1 l We will make repeated use of the following well-known Note that, p ; p 2 Fk ⊆ P and, therefore, ap < b C 1 holds for lemma: 2 f g C 1 ≤ f 0 1g C l 0; 1 . We derive ap 2 max ap ; ap < b 1 and thus 1 1 0 ap < b C . Since ap 2 Z, it follows that ap ≤ b, hence p 2 P . As Lemma 2.1 ([4, Lemma 6.33]). Let P be a rational polytope and let F 2 2 0 the choice of p 2 F C was arbitrary, we obtain F C ⊆ P . be a face of P. Then F 0 D P0 \ F. k 1 k 1 Note that, F ⊆ B . Thus, by Lemma 3.1, we have: n 2 n If P ⊆ T0; 1U and PI D;, Lemma 2.1 can be used to derive an upper bound on rk P . .k−2/ . / Corollary 3.2. Fk ⊆ Bn . Lemma 2.2 ([1, Lemma 3]). Let P ⊆ T0; 1Un be a polytope with The following theorem specifies a family of valid inequalities for .k/ B . PI D;. Then rk.P/ ≤ n. n n This bound is actually tight; a family of polytopes An ⊆ T0; 1U with Theorem 3.3. Let Bn be defined as above and k ≤ n. Then .An/I D; and rk.An/ D n was described in [3, p. 481]. X X xi C .1 − xi/ ≥ 1 For i 2 TnU, the i-th coordinate flip maps xi 7! 1−xi and xj 7! xj for i 6D j. Another property that we will extensively use is that i2I i2IQnI the Gomory–Chvátal operator is commutative with unimodular .k/ Q Q is valid for Bn for all I ⊆ I ⊆ TnU with jIj D n − k. Moreover, these transformations, in particular, coordinate flips. 1 inequalities can be derived as iterated 0; 2 -cuts. Lemma 2.3 ([5, Lemma 4.3]). Let P ⊆ T0; 1Un be a polytope and let Proof. The proof is by induction on k. First, let us look at the case u be a coordinate flip. Then .u.P//0 D u.P0/. D P C P − ≥ Q DT U k 0. By definition, i2I xi i2QInI .1 xi/ 1 with I n is valid for B . Now consider 0 < k ≤ n, and assume that the claim Given polytopes P ⊆ T0; 1Un; Q ⊆ T0; 1Uk, and a k-dimensional n Q Q n ∼ holds for k − 1. Let I ⊆ TnU with jIj D n − k be arbitrary.