The (not so) trivial lifting in two dimensions

Fukasawa, Ricardo; Poirrier, Laurent; Xavier, Álinson S.

doi:10.1007/s12532-018-0146-5

Cited by 8 publications

(5 citation statements)

References 22 publications

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“…Moreover, the problem of actually computing the cut coefficients is highly nontrivial for these "good" families from the literature (involving closest lattice vector problems, as discussed in point 2. above). The only family of sets in previous literature where more efficient algorithms exist to compute any lifting is the family of 2-dimensional b + Z n free convex sets and even there, it is ironically quite non-trivial to compute the trivial lifting [33]. But for the "good" family we propose above, even in arbitrary dimensions, we give an efficient algorithm to compute the trivial lifting (which also happens to be the unique minimal lifting).…”

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

“…In [32,33], the authors are explicitly concerned with computing the trivial lifting formula (1.3), without solving an integer linear program. In fact, our result outlined in Item 2 above is very much inspired by ideas from [33]. This, to the best of our knowledge, summarizes the most directly comparable literature on the efficiency aspect of cut generating functions.…”

Section: Introductionmentioning

confidence: 99%

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Can cut generating functions be good and efficient?

Basu¹

2018

Preprint

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Making cut generating functions (CGFs) computationally viable is a central question in modern integer programming research. One would like to find CGFs that are simultaneously good, i.e., there are good guarantees for the cutting planes they generate, and efficient, meaning that the values of the CGFs can be computed cheaply (with procedures that have some hope of being implemented in current solvers). We investigate in this paper to what extent this balance can be struck. We propose a family of CGFs which, in a sense, achieves this harmony between good and efficient. In particular, we provide a parameterized family of b+Z n free sets to derive CGFs from and show that our proposed CGFs give a good approximation of the closure given by CGFs obtained from all maximal b + Z n free sets and their so-called trivial liftings, and simultaneously, show that these CGFs can be computed with explicit, efficient procedures. We provide a constructive framework to identify these sets as well as computing their trivial lifting. We follow it up with computational experiments to demonstrate this and to evaluate their practical use. Our proposed family of cuts seem to give some tangible improvement on randomly generated instances compared to GMI cuts; however, in MIPLIB 3.0 instances, and vertex cover and stable problems on random graph instances, their performance is poor.

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Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Can cut generating functions be good and efficient?

Basu¹

2018

Preprint

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“…Lattice-free convex bodies appear in many important works concerning the theory of integer programming. They are central objects in cutting plane theory [1,6,9,5,2,8,10] and play a crucial role in Lenstra's algorithm [13] for integer programming in fixed dimension. A fundamental property of latticefree convex bodies is that they are "flat" with respect to the integer lattice.…”

Section: Introductionmentioning

confidence: 99%

Lattice-free simplices with lattice width $2d - o(d)$

Mayrhofer¹,

Jamico²,

Weltge³

2021

Preprint

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The Flatness theorem states that the maximum lattice width Flt(d) of a d-dimensional lattice-free convex set is finite. It is the key ingredient for Lenstra's algorithm for integer programming in fixed dimension, and much work has been done to obtain bounds on Flt(d). While most results have been concerned with upper bounds, only few techniques are known to obtain lower bounds. In fact, the previously best known lower bound Flt(d) ≥ 1.138d arises from direct sums of a 3-dimensional lattice-free simplex. In this work, we establish the lower bound Flt(d) ≥ 2d − O( √ d), attained by a family of lattice-free simplices. Our construction is based on a differential equation that naturally appears in this context. Additionally, we provide the first local maximizers of the lattice width of 4-and 5-dimensional lattice-free convex bodies.

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“…The elegance and convenience of generating cuts from the simplex tableau has motivated a recent stream of theoretical work on generating cuts from two rows of the simplex tableau (Andersen et al 2007, Cornuéjols andMargot 2008) and subsequently more rows (Borozan andCornuéjols 2009, Basu et al 2010), on how these cuts can be strengthened when nonbasic variables are integer (Dey and Wolsey 2010, Conforti et al 2011b, Basu et al 2013, Fukasawa et al 2016, and several other variants. The reader is referred to Conforti et al (2011a) and Basu et al (2015) for a broader review of this line of work, which has been accompanied by extensive computational experimentation (Espinoza 2008, Basu et al 2011, Dey et al 2014, Louveaux et al 2015.…”

Section: Introductionmentioning

confidence: 99%

When Lift-and-Project Cuts Are Different

Balas

Serra

2020

INFORMS Journal on Computing

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In this paper, we present a method to determine if a lift-and-project cut for a mixed-integer linear program is regular, in which case the cut is equivalent to an intersection cut from some basis of the linear relaxation. This is an important question due to the intense research activity for the past decade on cuts from multiple rows of simplex tableau as well as on lift-and-project cuts from non-split disjunctions. While it is known since Balas and Perregaard (2003) that such equivalence always holds for lift-and-project cuts from split disjunctions, Balas and Kis (2016) have recently shown that there is a necessary and sufficient condition in the case of arbitrary disjunctions: a lift-and-project cut is regular if, and only if, it corresponds to a regular solution of the Cut Generating Linear Program (CGLP). This paper has four contributions. First, we state a result that simplifies the verification of regularity for basic CGLP solutions from Balas and Kis (2016).Second, we provide a mixed-integer formulation that checks whether there is a regular CGLP solution for a given cut. Third, we describe a numerical procedure that verifies if a lift-and-project cut is not regular.Finally, we use this method to evaluate how often lift-and-project cuts are irregular, and thus not equivalent to multi-row cuts, on 74 instances of the MIPLIB benchmarks.

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The (not so) trivial lifting in two dimensions

Cited by 8 publications

References 22 publications

Can cut generating functions be good and efficient?

Can cut generating functions be good and efficient?

Lattice-free simplices with lattice width $2d - o(d)$

When Lift-and-Project Cuts Are Different

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