Clean Clutters and Dyadic Fractional Packings

Abdi, Ahmad; Cornuéjols, Gérard; Guenin, Bertrand; Tunçel, Levent

doi:10.1137/21m1397325

Cited by 5 publications

(20 citation statements)

References 31 publications

(50 reference statements)

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“…Then (D) has a dyadic optimal solution. This theorem confirms a conjecture of Seymour on ideal clutters, called the Dyadic Conjecture, for the clutter of T -joins (see [14], §79.3e, and [1]).…”

Section: (D)supporting

confidence: 85%

“…One must then revise the question, and may ask the following: Does (D) always have a quarterintegral optimal solution? This would then become a special case of the 1 4 -MFMC Conjecture for ideal clutters [6], and would follow from the generalized Fulkerson conjecture [16] (see also [5]).…”

Section: Introductionmentioning

confidence: 94%

“…A dyadic rational is an integer multiple of 1 2 k for some integer k ≥ 0. A vector is dyadic if each of its entries is a dyadic rational.…”

Section: (D)mentioning

confidence: 99%

See 2 more Smart Citations

On Dyadic Fractional Packings of $T$-Joins

Abdi¹,

Cornuéjols²,

Palion³

2022

SIAM J. Discrete Math.

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Let G = (V, E) be a graph, and T ⊆ V a nonempty subset of even cardinality. The famous theorem of Edmonds and Johnson on the T -join polyhedron implies that the minimum cardinality of a T -cut is equal to the maximum value of a fractional packing of T -joins. In this paper, we prove that the fractions assigned may be picked as dyadic rationals, i.e. of the form a 2 k for some integers a, k ≥ 0.

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Section: (D)supporting

confidence: 85%

Section: Introductionmentioning

confidence: 94%

See 1 more Smart Citation

On Dyadic Fractional Packings of $T$-Joins

Abdi¹,

Cornuéjols²,

Palion³

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%

“…It can be readily checked that the set of finitely p-adic numbers is the set of finite series of the form M i=N a i p i , where M, N ∈ Z, M ≥ N , and 0 ≤ a i < p, a i ∈ Z for all N ≤ i ≤ M , justifying our terminology. 2 In this paper, we will only deal with finitely p-adic numbers; for simplicity we refer to them as p-adic numbers throughout the paper. Also we refer to "2-adic" as "dyadic".…”

Section: Introductionmentioning

confidence: 99%

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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Clean Clutters and Dyadic Fractional Packings

Cited by 5 publications

References 31 publications

On Dyadic Fractional Packings of $T$-Joins

On Dyadic Fractional Packings of $T$-Joins

Total Dual Dyadicness and Dyadic Generating Sets

Total dual dyadicness and dyadic generating sets

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