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On the Unique-Lifting Property On the Unique-lifting Property Gennadiy Averkov1 and Amitabh Basu2 1 Institute of Mathematical Optimization, Faculty of Mathematics, University of Magdeburg 2 Department of Applied Mathematics and Statistics, The Johns Hopkins University Abstract. We study the uniqueness of minimal liftings of cut gener- ating functions obtained from maximal lattice-free polytopes. We prove a basic invariance property of unique minimal liftings for general max- imal lattice-free polytopes. This generalizes a previous result by Basu, Cornu´ejolsand K¨oppe [3] for simplicial maximal lattice-free polytopes, thus completely settling this fundamental question about lifting. We also extend results from [3] for minimal liftings in maximal lattice-free sim- plices to more general polytopes. These nontrivial generalizations require the use of deep theorems from discrete geometry and geometry of num- bers, such as the Venkov-Alexandrov-McMullen theorem on translative tilings, and McMullen's characterization of zonotopes. 1 Introduction Overview and Motivation. The idea of cut generating functions has emerged as a major theme in recent research on cutting planes for mixed-integer linear programming. The main object of study is the following family of mixed-integer sets: k ` n Xf (R; P ) = f(s; y) 2 R+ × Z+ : f + Rs + P y 2 Z g; where f 2 Rn, and R 2 Rn×k, P 2 Rn×`. We denote the columns of matrices R and P by ri, i = 1; : : : ; k and pj, j = 1; : : : ; ` respectively. For a fixed f 2 Rn nZn, a cut generating pair ( ; π) for f is a pair of functions ; π : Rn ! R such Pk i P` j that i=1 (r )si + j=1 π(p )yj ≥ 1 is a valid inequality (cutting plane) for Xf (R; P ), for all matrices R and P . This model and the idea of cut generating pairs arose in the work of Gomory and Johnson from the 70s. We refer the reader to [5] for a survey of the intense research activity this area has seen in the last 5-6 years. A very important class of cut generating pairs is obtained using the gauge function of maximal lattice-free polytopes in Rn. These are convex polytopes B ⊆ Rn such that int(B) \ Zn = ; and B is inclusion-wise maximal with this property. Given a maximal lattice-free polytope B such that f 2 int(B), one can express B = fx 2 Rn : ai · (x − f) ≤ 1 8i 2 Ig. One then obtains a cut i n generating pair for f by setting (r) = maxi2I a ·r for all r 2 R (this is known as the gauge of B −f), and using any π such that ( ; π) is a cut generating pair. n n One commonly used π is defined by π(r) = minw2Z (r + w) for all r 2 R . It can be shown that ( ; π) thus defined forms a valid cut generating pair. Given a particular maximal lattice-free polytope B with f 2 int(B), it is generally possible to find many different functions π such that ( ; π) is a cut generating pair for f, when is fixed to be the gauge of B − f. The different possible π's are called liftings and the strongest cutting planes are obtained from minimal liftings, i.e., π such that for every lifting π0, the inequality π0 ≤ π implies π0 = π. Closed-form formulas for cut generating pairs ( ; π) are highly desirable for computational purposes. This is the main motivation for considering the special class of cut generating pairs obtained from the gauge of maximal lattice-free polytopes and their liftings. The gauge is given by the very simple formula i (r) = maxi2I a · r, and the hope is that simple formulas can be found for its minimal liftings as well. In this regard, the following results are particularly useful. Dey and Wolsey established the following theorem in [6] for n = 2. Theorem 1. (Theorems 5 and 6 in [4], Theorem 4 in [2].) Let be the gauge of B − f, where B is a maximal lattice-free polytope and f 2 int(B). There exists a compact subset R0(f; B) ⊆ Rn such that R0(f; B) has nonempty interior, and for every minimal lifting π, π(r) = (r) if and only if r 2 R0(f; B). Moreover, for all minimal liftings π, π(r) = π(r + w) for every w 2 Zn, r 2 Rn. This theorem shows that for a \fat" periodic region R0(f; B) + Zn, we have a closed-form formula for all minimal liftings (using the formula for ). In particu- lar, when all the columns of the matrix P are in R0(f; B)+Zn, we can efficiently Pk i P` j find the cutting plane i=1 (r )si + j=1 π(p )yj ≥ 1 from B. Moreover, the above theorem shows that if R0(f; B) + Zn = Rn, then there is a unique min- imal lifting π. The following theorem from [2] establishes the necessity of this condition. Theorem 2. (Theorem 5 in [2].) Let be the gauge of B − f, where B is a maximal lattice-free polytope and f 2 int(B). Then has a unique minimal lifting if and only if R0(f; B)+Zn = Rn.(R0(f; B) is the region from Theorem 1) The above theorems provide a geometric perspective on sequence indepen- dent lifting and monoidal strengthening that started with the work of Balas and Jeroslow [1]. In this context, characterizing pairs f; B with unique minimal lift- ings becomes an important question in the cut generating function approach to cutting planes. Recent work and related literature can be found in [2{7]. Our Contributions We will denote the convex hull, affine hull, interior and rel- ative interior of a set X using conv(X); aff(X); int(X) and relint(X) respectively. n 1 2 We call an n-dimensional polytope S in R a spindle if S = (b + C1) \ (b + C2) 1 2 is the intersection of two translated polyhedral cones b + C1 and b + C2, such 1 2 that the apex b 2 int(b2 + C2) and the apex b 2 int(b1 + C1). Let B be a maximal lattice-free polytope in Rn and let f 2 int(B). By Fct(B) we denote the set of all facets of B. With each F and f we associate the set n PF (f) := conv(ffg [ F ). With each F and each z 2 F \ Z we associate the spindle SF;z(f) := PF (f) \ (z + f − PF (f)): Furthermore, we define RF (f) := S n SF;z(f), the union of all spindles arising from the facet F , and the z2F \Z S lifting region R(f; B) := F 2Fct(B) RF (f) associated with the point f. One of the main results in [2] was to establish that R(f; B) − f is precisely the region R0(f; B) described in Theorem 1. In light of Theorem 2, we say B has the unique-lifting property with respect to f if R(f; B) + Zn = Rn, and B has the multiple-lifting property with respect to f if R(f; B) + Zn 6= Rn. We summarize our main contributions in this paper. (i) A natural question arises: is it possible that B has the unique-lifting property with respect to one f1 2 int(B), and has the multiple-lifting property with respect to another f2 2 int(B)? This question was investigated in [3] and the main result was to establish that this cannot happen when B is a simplicial polytope. We prove this for general maximal lattice-free polytopes without the simpliciality assumption: Theorem 3. (Unique-lifting invariance theorem.) Let B be any maximal lattice-free polytope. For all f1; f2 2 int(B), B has the unique-lifting property with respect to f1 if and only if B has the unique-lifting property with respect to f2. In view of this result, we can speak about the unique-lifting property of B, without reference to any f 2 int(B). (ii) To prove Theorem 3, we first show that the volume of R(f; B)=Zn (the region R(f; B) sent onto the torus Rn=Zn) is an affine function of f (Theorem 4). This is also an extension of the corresponding theorem from [3] for simplicial B. Besides handling the general case, our proof is also significantly shorter and more elegant. We develop a tool for computing volumes on the torus, which enables us to circumvent a complicated inclusion-exclusion argument from [3] (see pages 349-350 in [3]). We view this volume computation tool as an important technical contribution of this paper. (iii) A major contribution of [3] was to characterize the unique-lifting property for a special class of simplices. We generalize all the results from [3] to a broader class of polytopes, called pyramids in Sections 3 and 5 (see Re- mark 1 and Theorems 5 and 11). For our generalizations, we build tools in Section 4 that invoke non-trivial theorems from the geometry of numbers and discrete geometry, such as the Venkov-Alexandrov-McMullen theorem for translative tilings in Rn, McMullen's characterizations of polytopes with centrally symmetric faces [9] and the combinatorial structure of zonotopes. (iv) Our techniques give an iterative procedure to construct new families of poly- topes with the unique-lifting property in every dimension n 2 N. This vastly expands the known list of polytopes with the unique-lifting property. See Remarks 1, 2 and 3. 2 Invariance Theorem on the Uniqueness of Lifting We consider the torus Tn = Rn=Zn, equipped with the natural Haar measure that assigns volume 1 to Rn=Zn.
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