On Dyadic Fractional Packings of $T$-Joins

Abdi, Ahmad; Cornuéjols, Gérard; Palion, Zuzanna

doi:10.1137/21m1445260

Cited by 4 publications

(9 citation statements)

References 13 publications

(23 reference statements)

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“…Both of these clutters are minimally non-ideal, that is, each clutter is non-ideal but every proper minor of it is ideal. It can be readily checked that for the first clutter, the joint optimal value of (P) and (D) is the non-dyadic number 7 3 , while for the second clutter this value is the non-dyadic number 8 3 . As L 7 is also a binary clutter, it follows that the analogue of Conjecture 1.1 does not hold for binary clutters.…”

Section: Possible Extensions and Restrictions Of Conjecture 11mentioning

confidence: 99%

“…Other than the clutter of T -cuts of a graph, this conjecture is known to hold for three other classes. Very recently, the conjecture was proved for the clutter of T -joins of a graph [7]. Given a signed graph, if the (binary) clutter of odd circuits is ideal, then the clutter must have an optimal fractional packing that is 1 2 -integral [21].…”

Section: Possible Extensions and Restrictions Of Conjecture 11mentioning

confidence: 99%

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Clean Clutters and Dyadic Fractional Packings

Abdi¹,

Cornuéjols²,

Guenin³

et al. 2022

SIAM J. Discrete Math.

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A vector is dyadic if each of its entries is a dyadic rational number, i.e., an integer multiple of 1 2 k for some nonnegative integer k. We prove that every clean clutter with a covering number of at least two has a dyadic fractional packing of value two. This result is best possible for there exist clean clutters with a covering number of three and no dyadic fractional packing of value three. Examples of clean clutters include ideal clutters, binary clutters, and clutters without an intersecting minor. Our proof is constructive and leads naturally to an albeit exponential algorithm. We improve the running time to quasi-polynomial in the rank of the input, and to polynomial in the binary case.

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Section: Possible Extensions and Restrictions Of Conjecture 11mentioning

confidence: 99%

Section: Possible Extensions and Restrictions Of Conjecture 11mentioning

confidence: 99%

Clean Clutters and Dyadic Fractional Packings

Abdi¹,

Cornuéjols²,

Guenin³

et al. 2022

SIAM J. Discrete Math.

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“…For example, when A is the T -cut versus edge incidence matrix of a graft, an optimal dual solution y is 1 2 -integral (Lovász [22]). Seymour's conjecture was proved recently in a couple of other special cases [2], [5].…”

Section: Introductionmentioning

confidence: 90%

“…We also deal with [p 0 ]-adic numbers, which are precisely the integers. 5 The guarantees we provide in this section will apply more generally to optimal solutions to the following [p k ]-adic linear program, for any integer k ≥ 0:…”

Section: Bounding the Support Size Of Dyadic Solutionsmentioning

confidence: 99%

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Total Dual Dyadicness and Dyadic Generating Sets

Abdi

Cornuéjols

Guenin

et al. 2022

Integer Programming and Combinatorial Optimization

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A rational number is dyadic if it has a finite binary representation p/2 k , where p is an integer and k is a nonnegative integer. Dyadic rationals are important for numerical computations because they have an exact representation in floating-point arithmetic on a computer. A vector is dyadic if all its entries are dyadic rationals. We study the problem of finding a dyadic optimal solution to a linear program, if one exists. We show how to solve dyadic linear programs in polynomial time. We give bounds on the size of the support of a solution as well as on the size of the denominators. We identify properties that make the solution of dyadic linear programs possible: closure under addition and negation, and density, and we extend the algorithmic framework beyond the dyadic case.

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Total dual dyadicness and dyadic generating sets

et al. 2023

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A vector is dyadic if each of its entries is a dyadic rational number, i.e. of the form a 2 k for some integers a, k with k ≥ 0. A linear system Ax ≤ b with integral data is totally dual dyadic if whenever min{b ⊺ y ∶ A ⊺ y = w, y ≥ 0} for w integral, has an optimal solution, it has a dyadic optimal solution. In this paper, we study total dual dyadicness, and give a co-NP characterization of it in terms of dyadic generating sets for cones and subspaces, the former being the dyadic analogue of Hilbert bases, and the latter a polynomial-time recognizable relaxation of the former. Along the way, we see some surprising turn of events when compared to total dual integrality, primarily led by the density of the dyadic rationals. Our study ultimately leads to a better understanding of total dual integrality and polyhedral integrality. We see examples from dyadic matrices, T -joins, circuits, and perfect matchings of a graph.

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On Dyadic Fractional Packings of $T$-Joins

Cited by 4 publications

References 13 publications

Clean Clutters and Dyadic Fractional Packings

Clean Clutters and Dyadic Fractional Packings

Total Dual Dyadicness and Dyadic Generating Sets

Total dual dyadicness and dyadic generating sets

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