The complexity of recognizing linear systems with certain integrality properties

Ding, Guoli; Li, Feng; Zang, Wenan

doi:10.1007/s10107-007-0103-y

Cited by 34 publications

(39 citation statements)

References 12 publications

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“…The method also gives a polynomial-time algorithm to tell whether a g-polymatroid is integral, see Theorem 4.16. In contrast, testing an arbitrary polyhedron for integrality [20] or TDI-ness is coNP-complete [5], the latter even for cones [19]. One might ask for a g-polymatroid P if it is true that every (p, b) such that Q(p, b) = P satisfies that (p, b) is TDL?…”

Section: Definition 12 (Tdl) the Pair (P B)mentioning

confidence: 99%

Characterizing and recognizing generalized polymatroids

Frank¹,

Király²,

Pap³

et al. 2013

Math. Program.

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Generalized polymatroids are a family of polyhedra with several nice properties and applications. One property of generalized polymatroids used widely in existing literature is "total dual laminarity;" we make this notion explicit and show that only generalized polymatroids have this property. Using this we give a polynomial-time algorithm to check whether a given linear program defines a generalized polymatroid, and whether it is integral if so. Additionally, whereas it is known that the intersection of two integral generalized polymatroids is integral, we show that no larger class of polyhedra satisfies this property.

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Section: Definition 12 (Tdl) the Pair (P B)mentioning

confidence: 99%

Characterizing and recognizing generalized polymatroids

Frank¹,

Király²,

Pap³

et al. 2013

Math. Program.

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“…Could this or some other "geometric" (polyhedral) method ever handle perfectness -a notion that can be defined in purely polyhedral terms -efficiently (in polynomial time) and in a less technical way ? In view of the NP-completeness of the general problem [4], [30], unimodular covering may provide a distinguishing clue for this 0 − 1 special case.…”

Section: Computationmentioning

confidence: 99%

“…The recognition of TDI systems has been recently proved to be coNP-complete by Ding, Feng and Zang [4] and this result has been sharpened to the recognition of explicitly given systems with only 0 − 1 coefficient vectors, and where the defined polyhedron has exactly one vertex (Hilbert basis testing by Pap [30]). Graph theory results of Chudnovsky, Cornuéjols, Liu, Seymour and Vušković [9] allow one to recognize TDI systems with 0 − 1 coefficient matrices and right hand sides.…”

Section: Introductionmentioning

confidence: 99%

Alternatives for testing total dual integrality

O'Shea

Sebő

2010

Math. Program.

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Abstract. In this paper we provide characterizing properties of 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.

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“…The question now arises, whether this property can also be tested in polynomial time, like it is the case for total unimodularity. Pap [13] recently proved that the recognition of Hilbert bases is coNPcomplete, see also [3]. Via the result of Giles and Orlin this also means that testing whether a tuple (A, c) has the integer rounding property is co-NP complete.…”

Section: Introductionmentioning

confidence: 99%

Testing additive integrality gaps

et al. 2012

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We consider the problem of testing whether the maximum additive integrality gap of a family of integer programs in standard form is bounded by a given constant. This can be viewed as a generalization of the integer rounding property, which can be tested in polynomial time if the number of constraints is fixed. It turns out that this generalization is NP-hard even if the number of constraints is fixed. However, if, in addition, the objective is the all-one vector, then one can test in polynomial time whether the additive gap is bounded by a constant.

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The complexity of recognizing linear systems with certain integrality properties

Cited by 34 publications

References 12 publications

Characterizing and recognizing generalized polymatroids

Characterizing and recognizing generalized polymatroids

Alternatives for testing total dual integrality

Testing additive integrality gaps

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