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Deciding Emptiness of the Gomory-Chvatal Closure is NP-Complete, Even for a Rational Polyhedron Containing No integer Point 18th

Deciding Emptiness of the Gomory-Chvatal Closure is NP-Compl...

引用

18th international conference on integer programming and combinatorial optimization (ipco)

作者： Cornuejols, Gerard Li, Yanjun Carnegie Mellon Univ Tepper Sch Business Pittsburgh PA 15213 USA Purdue Univ Krannert Sch Management W Lafayette IN 47906 USA

ISBN: (纸本)9783319334615;9783319334608

Gomory-Chvatal cuts are prominent in integer programming. the Gomory-Chvatal closure of a polyhedron is the intersection of all half spaces defined by its Gomory-Chvatal cuts. In this paper, we show that it is NP-complete to decide whether the Gomory-Chvatal closure of a rational polyhedron is empty, even when this polyhedron contains no integer point. this implies that the problem of deciding whether the Gomory-Chvatal closure of a rational polyhedron P is identical to the integer hull of P is NP-hard. Similar results are also proved for the {-1, 0, 1}-cuts and {0, 1}-cuts, two special types of Gomory-Chvatal cuts with coefficients restricted in {-1, 0, 1} and {0, 1}, respectively.

关键词： integer programming Gomory-Chvatal cuts Gomory-Chvatal closure integer hull Computational complexity

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Sub-Symmetry-Breaking Inequalities for ILP with Structured Symmetry 1

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20th international conference on integer programming and combinatorial optimization (ipco)

作者： Bendotti, Pascale Fouilhoux, Pierre Rottner, Cecile Sorbonne Univ CNRS 7606 LIP6 Paris France EDF R&D Palaiseau France

ISBN: (数字)9783030179533

ISBN: (纸本)9783030179533;9783030179526

We consider integer linear programs whose solutions are binary matrices and whose (sub-)symmetry groups are symmetric groups acting on (sub-)columns. We propose a framework to build (sub-)symmetry-breaking inequalities for such programs, by introducing one additional variable per sub-symmetry group considered. the proposed framework is applied to derive inequalities breaking both symmetries and sub-symmetries in the graph coloring problem and in the ramp constrained min-up/min-down unit commitment problem.

关键词： integer linear programming (Sub)-symmetry breaking inequalities Graph Coloring Problem Unit Commitment Problem

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On the Congruency-Constrained Matroid Base 25th

On the Congruency-Constrained Matroid Base

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25th international conference on integer programming and combinatorial optimization (ipco)

作者： Liu, Siyue Xu, Chao Univ Elect Sci & Technol China Chengdu Peoples R China Carnegie Mellon Univ Pittsburgh PA 15213 USA

ISBN: (纸本)9783031598340;9783031598357

Consider a matroid where all elements are labeled with an element in Z. We are interested in finding a base where the sum of the labels is congruent to g (mod m). We show that this problem can be solved in (O) over tilde (2(4m)nr(5/6)) time for a matroid with n elements and rank r, when m is either the product of two primes or a prime power. the algorithm can be generalized to all moduli and, in fact, to all abelian groups if a classic additive combinatorics conjecture by Schrijver and Seymour holds true. We also discuss the optimization version of the problem.

关键词： Matroid Additive Combinatorics optimization

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Sparsity and Integrality Gap Transference Bounds for integer Programs 25th

Sparsity and Integrality Gap Transference Bounds for Integer...

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25th international conference on integer programming and combinatorial optimization (ipco)

作者： Aliev, Iskander Celaya, Marcel Henk, Martin Cardiff Univ Cardiff Wales Tech Univ Berlin Berlin Germany

ISBN: (纸本)9783031598340;9783031598357

We obtain new transference bounds that connect two active areas of research: proximity and sparsity of solutions to integer programs. Specifically, we study the additive integrality gap of the integer linear programs min{c.x : x is an element of P boolean AND Z(n)}, where P = {x is an element of R-n : Ax = b, x >= 0} is a polyhedron in the standard form determined by an integer m x n matrix A and an integer vector b. the main result of the paper gives an upper bound for the integrality gap that drops exponentially in the size of support of the optimal solutions corresponding to the vertices of the integer hull of P. Additionally, we obtain a new proximity bound that estimates the l(2)-distance from a vertex of P to its nearest integer point in the polyhedron P. the proofs make use of the results from the geometry of numbers and convex geometry.

关键词： Integrality gap Proximity Sparsity

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A Lower Bound for the Max Entropy Algorithm for TSP 25th

A Lower Bound for the Max Entropy Algorithm for TSP

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25th international conference on integer programming and combinatorial optimization (ipco)

作者： Jin, Billy Klein, Nathan Williamson, David P. Cornell Univ Ithaca NY 14850 USA Inst Adv Study Olden Lane Princeton NJ 08540 USA

ISBN: (纸本)9783031598340;9783031598357

One of the most famous conjectures in combinatorial optimization is the four-thirds conjecture, which states that the integrality gap of the subtour LP relaxation of the TSP is equal to 4/3. For 40 years, the best known upper bound was 1.5, due to Wolsey [Wol80]. Recently, Karlin, Klein, and Oveis Gharan [KKO22] showed that the max entropy algorithm for the TSP gives an improved bound of 1.5 - 10(-36). In this paper, we show that the approximation ratio of the max entropy algorithm is at least 1.375, even for graphic TSP. thus the max entropy algorithm does not appear to be the algorithm that will ultimately resolve the four-thirds conjecture in the affirmative, should that be possible.

关键词： combinatorial optimization

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Persistency of Linear programming Relaxations for the Stable Set Problem 21st

Persistency of Linear Programming Relaxations for the Stable...

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21st international conference on integer programming and combinatorial optimization (ipco)

作者： Rodriguez-Heck, Elisabeth Stickler, Karl Walter, Matthias Weltge, Stefan Rhein Westfal TH Aachen Lehrstuhl Operat Res Aachen Germany Univ Twente Dept Appl Math Enschede Netherlands Tech Univ Munich Dept Math Munich Germany

ISBN: (纸本)9783030457716;9783030457709

the Nemhauser-Trotter theorem states that the standard linear programming (LP) formulation for the stable set problem has a remarkable property, also known as (weak) persistency: for every optimal LP solution that assigns integer values to some variables, there exists an optimal integer solution in which these variables retain the same values. While the standard LP is defined by only non-negativity and edge constraints, a variety of stronger LP formulations have been studied and one may wonder whether any of them has the this property as well. We show that any stronger LP formulation that satisfies mild conditions cannot have the persistency property on all graphs, unless it is always equal to the stable-set polytope.

关键词： Persistency integer linear programming Stable set

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Intersection Cuts for Factorable MINLP 20th

Intersection Cuts for Factorable MINLP

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20th international conference on integer programming and combinatorial optimization (ipco)

作者： Serrano, Felipe Zuse Inst Berlin Optimizat Dept Takustr 7 D-14195 Berlin Germany

ISBN: (纸本)9783030179533;9783030179526

Given a factorable function f, we propose a procedure that constructs a concave underestimator of f that is tight at a given point. these underestimators can be used to generate intersection cuts. A peculiarity of these underestimators is that they do not rely on a bounded domain. We propose a strengthening procedure for the intersection cuts that exploits the bounds of the domain. Finally, we propose an extension of monoidal strengthening to take advantage of the integrality of the non-basic variables.

关键词： Mixed-integer nonlinear programming Intersection cuts Monoidal strengthening

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Linear programming Using Limited-Precision Oracles 20th

Linear Programming Using Limited-Precision Oracles

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20th international conference on integer programming and combinatorial optimization (ipco)

作者： Gleixner, Ambros Steffy, Daniel E. Konrad Zuse Zentrum Informat Tech Berlin Takustr 7 D-14195 Berlin Germany Oakland Univ Math & Stat Rochester MI 48063 USA

ISBN: (纸本)9783030179533;9783030179526

Linear programming is a foundational tool for many aspects of integer and combinatorial optimization. this work studies the complexity of solving linear programs exactly over the rational numbers through use of an oracle capable of returning limited-precision LP solutions. Under mild assumptions, it is shown that a polynomial number of calls to such an oracle and a polynomial number of bit operations, is sufficient to compute an exact solution to an LP. Previous work has often considered oracles that provide solutions of an arbitrary specified precision. While this leads to polynomial-time algorithms, the level of precision required is often unrealistic for practical computation. In contrast, our work provides a foundation for understanding and analyzing the behavior of the methods that are currently most effective in practice for solving LPs exactly.

关键词： Linear programming Oracle complexity Diophantine approximation Exact solutions Symbolic computation Rational arithmetic Extended-precision arithmetic Iterative refinement

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Fast combinatorial Algorithms for Efficient Sortation 25th

Fast Combinatorial Algorithms for Efficient Sortation

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25th international conference on integer programming and combinatorial optimization (ipco)

作者： Van Dyk, Madison Klause, Kim Koenemann, Jochen Megow, Nicole Univ Waterloo Dept Combinator & Optimizat Waterloo ON Canada Univ Bremen Fac Math & Comp Sci Bremen Germany Amazon Modeling & Optimizat Seattle WA USA

ISBN: (纸本)9783031598340;9783031598357

Modern parcel logistic networks are designed to ship demand between given origin, destination pairs of nodes in an underlying directed network. Efficiency dictates that volume needs to be consolidated at intermediate nodes in a typical hub-and-spoke fashion. In practice, such consolidation requires parcel sortation. In this work, we propose a mathematical model for the physical requirements, and limitations of parcel sortation. We then show that it is NP-hard to determine whether a feasible sortation plan exists. We discuss several settings, where (near-) feasibility of a given sortation instance can be determined efficiently. the algorithms we propose are fast and build on combinatorial witness set type lower bounds that are reminiscent and extend those used in earlier work on degree-bounded spanning trees and arborescences.

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Rescaled Coordinate Descent Methods for Linear programming 1

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18th international conference on integer programming and combinatorial optimization (ipco)

作者： Dadush, Daniel Vegh, Laszlo A. Zambelli, Giacomo Ctr Wiskunde & Informat Amsterdam Netherlands London Sch Econ London England

ISBN: (数字)9783319334615

ISBN: (纸本)9783319334615;9783319334608

We propose two simple polynomial-time algorithms to find a positive solution to Ax = 0. Both algorithms iterate between coordinate descent steps similar to von Neumann's algorithm, and rescaling steps. In both cases, either the updating step leads to a substantial decrease in the norm, or we can infer that the condition measure is small and rescale in order to improve the geometry. We also show how the algorithms can be extended to find a solution of maximum support for the system Ax = 0, x >= 0. this is an extended abstract. the missing proofs will be provided in the full version.

关键词： Linear programming

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