Total dual integrality and b-matchings

Schrijver has shown that any rational polyhedron is the solution set of a unique minimal integer TDI linear system. We characterize this system for the case of b-matchings, one of the few known cases for which such a system is strictly larger than a minimal linear system sufficient to define the polyhedron.     

Total dual integrality of the linear complementarity problem

Annals of Operations Research - In this paper, we introduce total dual integrality of the linear complementarity problem (LCP) by analogy with the linear programming problem. The main idea of...     

On recognizing integer polyhedra

We give a simple proof that, determining whether a convex polytope has a fractional vertex, is NP-complete.Research supported by the National Science Foundation.     

Box-total dual integrality and edge-connectivity

Mathematical Programming - Given a graph $$G=(V,E)$$ and an integer $$k\ge 1$$ , the graph $$H=(V,F)$$ , where F is a family of elements (with repetitions allowed) of E, is a k-edge-connected...     

On integer points in polyhedra

We give an upper bound on the number of vertices ofP _I , the integer hull of a polyhedronP, in terms of the dimensionn of the space, the numberm of inequalities required to describeP, and the size of these inequalities. For fixedn the bound isO(m ^{n n–} ). We also describe an algorithm which determines the number of integer points in a polyhedron to within a multiplicative factor of 1+ in time polynomial inm, and 1/ when the dimensionn is fixed.Supported by Sonderfschungsbereich 303 (DFG) and NSF grant ECS-8611841.Partially supported by NSF grant DMS-8905645.Supported by NSF grants ECS-8418392 and CCR-8805199.mcd%vax.oxford.ac.uk     

On total dual integrality

We prove that each (rational) polyhedron of full dimension is determined by a unique minimal total dual integral system of linear inequalities, with integral left hand sides (thus extending a result of Giles and Pulleyblank), and we give a characterization of total dual integrality.     

The integrality number of an integer program

Mathematical Programming - We introduce the integrality number of an integer program (IP). Roughly speaking, the integrality number is the smallest number of integer constraints needed to solve an...     

Operations that preserve total dual integrality

There are many useful operations, such as adding slack variables, taking scalar multiples of inequalities, and applying Fourier-Motzkin elimination, that can be performed on a linear system such that if the system defines an integer polyhedron then so does the derived system. The topic dealt with here is whether or not these operations also preserve total dual integrality of linear systems.     

Alternatives for testing total dual integrality

In this paper we provide characterizing properties of totally dual integral (TDI) systems, among others the following: a system of linear inequalities is TDI if and only if its coefficient vectors form a Hilbert basis, and there exists a test-set for the system’s dual integer programs where all test vectors have positive entries equal to 1. Reformulations of this provide relations between computational algebra and integer programming and they contain Applegate, Cook and McCormick’s sufficient condition for the TDI property and Sturmfels’ theorem relating toric initial ideals generated by square-free monomials to unimodular triangulations. We also study the theoretical and practical efficiency and limits of the characterizations of the TDI property presented here. In the particular case of set packing polyhedra our results correspond to endowing the weak perfect graph theorem with an additional, computationally interesting, geometric feature: the normal fan of the stable set polytope of a perfect graph can be refined into a regular triangulation consisting only of unimodular cones.     

The rank of (mixed-) integer polyhedra

We define a purely geometrical notion of the rank of (mixed-) integer rational polyhedra that differs substantially from the existing notions found in the literature.     

Relaxations of mixed integer sets from lattice-free polyhedra

This paper gives an introduction to a recently established link between the geometry of numbers and mixed integer optimization. The main focus is to provide a review of families of lattice-free polyhedra and their use in a disjunctive programming approach. The use of lattice-free polyhedra in the context of deriving and explaining cutting planes for mixed integer programs is not only mathematically interesting, but it leads to some fundamental new discoveries, such as an understanding under which conditions cutting planes algorithms converge finitely.     

Box-total dual integrality,box-integrality,and equimodular matrices

Mathematical Programming - Box-totally dual integral (box-TDI) polyhedra are polyhedra described by systems which yield strong min-max relations. We characterize them in several ways, involving the...     

On integer points in polyhedra: A lower bound

Given a polyhedronP we writeP _I for the convex hull of the integral points inP. It is known thatP _I can have at most135-2 vertices ifP is a rational polyhedron with size . Here we give an example showing thatP _I can have as many as ( ^n–1) vertices. The construction uses the Dirichlet unit theorem.The results of the paper were obtained while this author was visiting the Cowles Foundation at Yale University     

On box totally dual integral polyhedra

Edmonds and Giles introduced the class of box totally dual integral polyhedra as a generalization of submodular flow polyhedra. In this paper a geometric characterization of these polyhedra is given. This geometric result is used to show that each TDI defining system for a box TDI polyhedron is in fact a box TDI system, that the class of box TDI polyhedra is in co-NP and is closed under taking projections and dominants, that the class of box perfect graphs is in co-NP, and a result of Edmonds and Giles which is related to the facets of box TDI polyhdera.Supported by a grant from the Alexander von Humboldt-Stiftung.     

Improved integer programming bounds using intersections of corner polyhedra

Consider the relaxation of an integer programming (IP) problem in which the feasible region is replaced by the intersection of the linear programming (LP) feasible region and the corner polyhedron for a particular LP basis. Recently a primal-dual ascent algorithm has been given for solving this relaxation. Given an optimal solution of this relaxation, we state criteria for selecting a new LP basis for which the associated relaxation is stronger. These criteria may be successively applied to obtain either an optimal IP solution or a lower bound on the cost of such a solution. Conditions are given for equality of the convex hull of feasible IP solutions and the intersection of all corner polyhedra.     

Integral decomposition of polyhedra and some applications in mixed integer programming

 This paper addresses the question of decomposing an infinite family of rational polyhedra in an integer fashion. It is shown that there is a finite subset of this family that generates the entire family. Moreover, an integer analogue of Carathéodory's theorem carries over to this general setting. The integer decomposition of a family of polyhedra has some applications in integer and mixed integer programming, including a test set approach to mixed integer programming. Received: May 22, 2000 / Accepted: March 19, 2002 Published online: December 19, 2002 Key words. mixed integer programming – test sets – indecomposable polyhedra – Hilbert bases – rational polyhedral cones This work was supported partially by the DFG through grant WE1462, by the Kultusministerium of Sachsen Anhalt through the grants FKZ37KD0099 and FKZ 2945A/0028G and by the EU Donet project ERB FMRX-CT98-0202. The first named author acknowledges the hospitality of the International Erwin Schr?dinger Institute for Mathematical Physics in Vienna, where a main part of his contribution to this work has been completed. Mathematics Subject Classification (1991): 52C17, 11H31