Split Closure Research Articles

On polytopes with linear rank with respect to generalizations of the split closure

In this paper we study the rank of polytopes contained in the 0-1 cube with respect to t-branch split cuts and t-dimensional lattice cuts for a fixed positive integer t. These inequalities are the same as split cuts when t=1 and generalize split cuts when t>1. For polytopes contained in the n-dimensional 0-1 cube, the work of Balas implies that the split rank can be at most n, and this bound is tight as Cornuéjols and Li gave an example with split rank n. All known examples with high split rank – i.e., at least cn for some positive constant c<1 – are defined by exponentially many (as a function of n) linear inequalities. For any fixed integer t>0, we give a family of polytopes contained in [0,1]n for sufficiently large n such that each polytope has empty integer hull, is defined by O(n) inequalities, and has rank Ω(n) with respect to t-dimensional lattice cuts. Therefore the split rank of these polytopes is Ω(n). It was shown earlier that there exist generalized branch-and-bound proofs, with logarithmic depth, of the nonexistence of integer points in these polytopes. Therefore, our lower bound results on split rank show an exponential separation between the depth of branch-and-bound proofs and split rank.

Reconstruction of Full Thickness Scalp Defects in Extremely Elderly Patients Using Dermal Regeneration Templates.

The reconstruction of full thickness scalp defects following oncologic resection poses a unique challenge that is further magnified in "extremely elderly" patients, defined as those in at least their ninth decade of life, who are often unsuitable candidates for complex reconstruction. A "simpler" option is two-stage reconstruction: placement of Integra dermal regenerative template (Integra Life Science, Plainsboro, NJ) followed by a split thickness skin graft (STSG). This case series illustrates the success of this technique in the extremely elderly. A retrospective analysis of patients > 80 years at the time of surgery who underwent full thickness scalp reconstruction following tumor extirpation in a two-stage approach under the care of single surgeon from January 2010 to June 2019 was conducted. Variables reviewed were medical history, surgical treatment response, time to split thickness skin graft, follow up, and success of wound coverage. Fourteen patients, with a mean age of 87 years (range: 80 to 101, median: 87), met inclusion criteria. Split thickness skin grafts were placed after an average of 18 days. Twelve patients had successful two-stage reconstruction with 100% take. One patient developed a hematoma under a portion of the template that neither required reoperation nor delayed split thickness skin graft placement. A second suffered from insufficient vascularization of the template with delay to split thickness skin graft and incomplete wound closure. This two-stage approach is a successful primary reconstructive option for definitive management of full thickness scalp defects following oncologic resection in extremely elderly patients.

Lattice closures of polyhedra

Given \(P\subset {\mathbb {R}}^n\), a mixed-integer set \(P^I=P\cap ({\mathbb {Z}}^{t}\times {\mathbb {R}}^{n-t}\)), and a k-tuple of n-dimensional integral vectors \((\pi _1, \ldots , \pi _k)\) where the last \(n-t\) entries of each vector is zero, we consider the relaxation of \(P^I\) obtained by taking the convex hull of points x in P for which \( \pi _1^Tx,\ldots ,\pi ^T_kx\) are integral. We then define the k-dimensional lattice closure of \(P^I\) to be the intersection of all such relaxations obtained from k-tuples of n-dimensional vectors. When P is a rational polyhedron, we show that given any collection of such k-tuples, there is a finite subcollection that gives the same closure; more generally, we show that any k-tuple is dominated by another k-tuple coming from the finite subcollection. The k-dimensional lattice closure contains the convex hull of \(P^I\) and is equal to the split closure when \(k=1\). Therefore, a result of Cook et al. (Math Program 47:155–174, 1990) implies that when P is a rational polyhedron, the k-dimensional lattice closure is a polyhedron for \(k=1\) and our finiteness result extends this to all \(k\ge 2\). We also construct a polyhedral mixed-integer set with n integer variables and one continuous variable such that for any \(k < n\), finitely many iterations of the k-dimensional lattice closure do not give the convex hull of the set. Our result implies that t-branch split cuts cannot give the convex hull of the set, nor can valid inequalities from unbounded, full-dimensional, convex lattice-free sets.

Lower Bounds on the Lattice-Free Rank for Packing and Covering Integer Programs

In this paper, we present lower bounds on the rank of the split closure, the multi-branch closure, and the lattice-free closure for packing sets as a function of the integrality gap. We also provid...

On the NP-hardness of deciding emptiness of the split closure of a rational polytope in the 0,1 hypercube

Split cuts are prominent general-purpose cutting planes in integer programming. The split closure of a rational polyhedron is what is obtained after intersecting the half-spaces defined by all the split cuts for the polyhedron. In this paper, we prove that deciding whether the split closure of a rational polytope is empty is NP-hard, even when the polytope is contained in the unit hypercube. As a direct corollary, we prove that optimization and separation over the split closure of a rational polytope in the unit hypercube are NP-hard, extending an earlier result of Caprara and Letchford.

Cutting planes from extended LP formulations

Given a mixed-integer set defined by linear inequalities and integrality requirements on some of the variables, we consider extended formulations of its continuous (LP) relaxation and study the effect of adding cutting planes in the extended space. In terms of optimization, extended LP formulations do not lead to better bounds as their projection onto the original space is precisely the original LP relaxation. However, adding cutting planes in the extended space can lead to stronger bounds. In this paper we show that for every 0---1 mixed-integer set with n integer and k continuous variables, there is an extended LP formulation with $$2n+k-1$$2n+k-1 variables whose elementary split closure is integral. The proof is constructive but it requires an inner description of the LP relaxation. We then extend this idea to general mixed-integer sets and construct the best extended LP formulation for such sets with respect to lattice-free cuts. We also present computational results on the two-row continuous group relaxation showing the strength of cutting planes derived from extended LP formulations.

On the polyhedrality of cross and quadrilateral closures

Split cuts form a well-known class of valid inequalities for mixed-integer programming problems. Cook et al. (Math Program 47:155---174, 1990) showed that the split closure of a rational polyhedron P is again a polyhedron. In this paper, we extend this result from a single rational polyhedron to the union of a finite number of rational polyhedra. We then use this result to prove that cross cuts yield closures that are rational polyhedra. Cross cuts are a generalization of split cuts introduced by Dash et al. (Math Program 135:221---254, 2012). Finally, we show that the quadrilateral closure of the two-row continuous group relaxation is a polyhedron, answering an open question in Basu et al. (Math Program 126:281---314, 2011).

Cut generation through binarization

For a mixed integer linear program where all integer variables are bounded, we study a reformulation introduced by Roy that maps general integer variables to a collection of binary variables. We study theoretical properties and empirical strength of rank-2 simple split cuts of the reformulation. We show that for a pure integer problem with two integer variables, these cuts are sufficient to obtain the integer hull of the problem, but that this does not generalize to problems in higher dimensions. We also give an algorithm to compute an approximation of the rank-2 simple split cut closure. We report empirical results on 22 benchmark instances showing that the bounds obtained compare favorably with those obtained with other approximate methods to compute the split closure or lattice-free cut closure. It gives a better bound than the split closure on 6 instances while it is weaker on 9 instances, for an average gap closed 3.8 % smaller than the one for the split closure.

Characterization of the split closure via geometric lifting

We analyze split cuts from the perspective of cut generating functions via geometric lifting. We show that $\alpha$-cuts, a natural higher-dimensional generalization of the $k$-cuts of Cornu\'{e}jols et al., gives all the split cuts for the mixed-integer corner relaxation. As an immediate consequence we obtain that the $k$-cuts are equivalent to split cuts for the 1-row mixed-integer relaxation. Further, we show that split cuts for finite-dimensional corner relaxations are restrictions of split cuts for the infinite-dimensional relaxation. In a final application of this equivalence, we exhibit a family of pure-integer programs whose split closures have arbitrarily bad integrality gap. This complements the mixed-integer example provided by Basu et al [On the relative strength of split, triangle and quadrilateral cuts, Math. Program. 126(2):281--314, 2011].

The surgical management of angiosarcoma of the breast—a single institution experience

Angiosarcoma of the breast is a rare and aggressive tumour of the vascular endothelium. It may arise spontaneously or secondary to radiation. We present our experience of managing breast angiosarcoma, the largest single institution case series from the UK to date along with a summary of the relevant literature. Data on all patients with breast angiosarcoma treated in our unit were prospectively recorded (2002–2014). Demographics, surgical details and outcomes were analysed. Eighteen female patients presented with breast angiosarcoma. Sixteen patients previously underwent adjuvant radiotherapy following surgery for breast carcinoma; the mean duration between radiotherapy and angiosarcoma development was 8.4 years (range 3–21). Resections were as follows: radical mastectomy (n = 14), simple mastectomy (n = 1) and wide local excision (n = 3). Reconstruction was undertaken as follows: pedicled lattisimus dorsi (LD) musculocutaneous flap (n = 5), pedicled LD muscle flap and split skin graft (n = 7), free deep inferior epigastric perforator (DIEP) flap (n = 1), pedicled vertical rectus abdominus muscle (VRAM) flap and split skin graft (n = 1), pedicled LD muscle and pedicled VRAM muscle flaps and split skin graft (n = 1), pedicled LD muscle and pedicled (contralateral) pectoralis major muscle flaps and split skin graft (n = 1) and direct closure (n = 2). Three patients developed local recurrence; mean duration from resection to recurrence was 12 months (range 9–19). Three patients developed metastasis. Seven patients (38.8 %) died; median survival from presentation was 19 months (range 2–55 months). The remaining eleven patients remain well with no disease recurrence; mean follow-up was 38 months (range 4–125). The estimated 5-year survival (Kaplan-Meier equation) in our cohort was 49 %. Breast angiosarcomas are rare and challenging to manage. Successful outcomes can be achieved by early, aggressive resection and appropriate reconstruction. Level of Evidence: Level IV, therapeutic study.

Approximating the Split Closure

The split closure has been proved in practice to be a very tight approximation of the integer hull formulation of a generic mixed-integer linear program. However, exact separation procedures for optimizing over the split closure have unacceptable computing times in practice; hence, many different heuristic strategies have been proposed in the last few years. In this paper we present a new overall framework for approximating the split closure that merges different ideas from the previous approaches. Computational results prove the effectiveness of the proposed procedure compared to the state of the art, showing that a good approximation of the split closure bound can be obtained with very reasonable computing times.

The triangle closure is a polyhedron

Recently, cutting planes derived from maximal lattice-free convex sets have been studied intensively by the integer programming community. An important question in this research area has been to decide whether the closures associated with certain families of lattice-free sets are polyhedra. For a long time, the only result known was the celebrated theorem of Cook, Kannan and Schrijver who showed that the split closure is a polyhedron. Although some fairly general results were obtained by Andersen et al. (Math Oper Res 35(1):233---256, 2010) and Averkov (Discret Optimiz 9(4):209---215, 2012), some basic questions have remained unresolved. For example, maximal lattice-free triangles are the natural family to study beyond the family of splits and it has been a standing open problem to decide whether the triangle closure is a polyhedron. In this paper, we show that when the number of integer variables $$m=2$$ the triangle closure is indeed a polyhedron and its number of facets can be bounded by a polynomial in the size of the input data. The techniques of this proof are also used to give a refinement of necessary conditions for valid inequalities being facet-defining due to Cornuejols and Margot (Math Program 120:429---456, 2009) and obtain polynomial complexity results about the mixed integer hull.

Experiments with Two-Row Cuts from Degenerate Tableaux

There has been a recent interest in cutting planes generated from two or more rows of the optimal simplex tableau. One can construct examples of integer programs for which a single cutting plane generated from two rows dominates the entire split closure. Motivated by these theoretical results, we study the effect of adding a family of cutting planes generated from two rows on a set of instances from the MIPLIB library. The conclusion of whether these cuts are competitive with Gomory mixed-integer cuts is very sensitive to the experimental setup. In particular, we consider the issue of reliability versus aggressiveness of the cut generators, an issue that is usually not addressed in the literature.

A probabilistic comparison of the strength of split, triangle, and quadrilateral cuts

We consider mixed integer linear sets defined by two equations involving two integer variables and any number of non-negative continuous variables. We analyze the benefit from adding a non-split inequality on top of the split closure. Applying a probabilistic model, we show that the importance of a type 2 triangle inequality decreases with decreasing lattice width, on average. Our results suggest that this is also true for type 3 triangle and quadrilateral inequalities.

The split closure of a strictly convex body

The Chvátal–Gomory closure and the split closure of a rational polyhedron are rational polyhedra. It has been recently shown that the Chvátal–Gomory closure of a strictly convex body is also a rational polytope. In this note, we show that the split closure of a strictly convex body is defined by a finite number of split disjunctions, but is not necessarily polyhedral. We also give a closed form expression in the original variable space of a split cut for full-dimensional ellipsoids.

An Analysis of Mixed Integer Linear Sets Based on Lattice Point Free Convex Sets

A maximal lattice free polyhedron L has max-facet-width equal to w if [Formula: see text] for all facets [Formula: see text] of L, and [Formula: see text] for some facet [Formula: see text] of L. The set obtained by adding all cuts whose validity follows from a maximal lattice free polyhedron with max-facet-width at most w is called the wth split closure. We show the wth split closure is a polyhedron. This generalizes a previous result showing this to be true when w = 1. We also consider the design of finite cutting plane proofs for the validity of an inequality. Given a measure of “size” of a maximal lattice free polyhedron, a natural question is how large a size s* of a maximal lattice free polyhedron is required to design a finite cutting plane proof for the validity of an inequality. We characterize s* based on the faces of the linear relaxation of the mixed integer linear set.

The Chvátal closure of generalized stable sets in bidirected graphs

Abstract We consider the set of integral solutions of A x ⩽ b , x ⩾ 0 , where A is the edge-vertex incidence matrix of a bidirected graph. We characterize its corner polyhedron, i.e. the convex hull of the points satisfying all the constraints except the non-negativity of the basic variables. We show that the non-trivial inequalities necessary to describe this polyhedron can be derived as fractional Gomory cuts. It follows in particular that the split closure is equal to the Chvatal closure in this case.

Relations between facets of low- and high-dimensional group problems

One-dimensional infinite group problems have been extensively studied and have yielded strong cutting planes for mixed integer programs. Although numerical and theoretical studies suggest that group cuts can be significantly improved by considering higher-dimensional groups, there are no known facets for infinite group problems whose dimension is larger than two. In this paper, we introduce an operation that we call sequential-merge. We prove that the sequential-merge operator creates a very large family of facet-defining inequalities for high-dimensional infinite group problems using facet-defining inequalities of lower-dimensional group problems. Further, they exhibit two properties that reflect the benefits of using facets of high-dimensional group problems: they have continuous variables’ coefficients that are not dominated by those of the constituent low-dimensional cuts and they can produce cutting planes that do not belong to the first split closure of MIPs. Further, we introduce a general scheme for generating valid inequalities for lower-dimensional group problems using valid inequalities of higher-dimensional group problems. We present conditions under which this construction generates facet-defining inequalities when applied to sequential-merge inequalities. We show that this procedure yields some two-step MIR inequalities of Dash and Gunluk.

Stable sets, corner polyhedra and the Chvátal closure

We consider the edge formulation of the stable set problem. We characterize its corner polyhedron, i.e. the convex hull of the points satisfying all the constraints except the non-negativity of the basic variables. We show that the non-trivial inequalities necessary to describe this polyhedron can be derived from one row of the simplex tableau as fractional Gomory cuts. It follows that the split closure is not stronger than the Chvátal closure for the edge relaxation of the stable set problem.

Cook, Kannan and Schrijver’s example revisited

In 1990, Cook, Kannan and Schrijver [W. Cook, R. Kannan, A. Schrijver, Chvátal closures for mixed integer programming problems, Mathematical Programming 47 (1990) 155–174] proved that the split closure (the 1st 1-branch split closure) of a polyhedron is again a polyhedron. They also gave an example of a mixed-integer polytope in R 2 + 1 whose 1-branch split rank is infinite. We generalize this example to a family of high-dimensional polytopes and present a closed-form description of the k th 1-branch split closure of these polytopes for any k ≥ 1 . Despite the fact that the m -branch split rank of the ( m + 1 )-dimensional polytope in this family is 1, we show that the 2-branch split rank of the ( m + 1 )-dimensional polytope is infinite when m ≥ 3 . We conjecture that the t -branch split rank of the ( m + 1 )-dimensional polytope of the family is infinite for any 1 ≤ t ≤ m − 1 and m ≥ 2 .