Consequences of the packing problem

Bodas, Hrishikesh; Drabkin, Benjamin; Fong, Caleb; Jin, S.; Kim, Justin; Li, Wenxuan; Seceleanu, Alexandra; Tang, Tingting; Williams, B. A.

doi:10.1007/s10801-021-01039-5

Cited by 4 publications

(3 citation statements)

References 16 publications

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“…As a consequence of Theorem 3.4 we recover the fact that α(I) can be realized as the value of the optimal solution of a linear program [4,Theorem 3.2]. If A is the incidence matrix of I and Q(A) is integral, we recover the following formulas [6,Theorem 4. where α(I) be the least degree of a minimal generator of I (Corollary 3.8).…”

Section: Introductionmentioning

confidence: 86%

“…The equality between symbolic and ordinary powers of squarefree monomial ideals was related to a conjecture of Conforti and Cornuéjols [12,Conjecture 1.6] on the max-flow min-cut property of clutters in [19,Theorem 4.6,Conjecture 4.18] and [20,Conjecture 3.10]. The Conforti and Cornuéjols conjecture is known in the context of symbolic powers as the Packing Problem [1,6,13,17,25,31] and it is a central problem in this theory.…”

Section: Introductionmentioning

confidence: 99%

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Rees algebras of filtrations of covering polyhedra and integral closure of powers of monomial ideals

Grisalde¹,

Seceleanu²,

Villarreal³

2021

Preprint

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The aims of this work are to study Rees algebras of filtrations of monomial ideals associated to covering polyhedra of rational matrices with non-negative entries and non-zero columns using combinatorial optimization and integer programming, and to study powers of monomial ideals and their integral closures using irreducible representations and polyhedral geometry. We study the Waldschmidt constant and the ic-resurgence of the filtration associated to a covering polyhedron and show how to compute these constants using linear programming. Then we show lower bounds for the ic-resurgence of the ideal of covers of a graph and prove that the lower bound is attained when the graph is perfect.

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Section: Introductionmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

Rees algebras of filtrations of covering polyhedra and integral closure of powers of monomial ideals

Grisalde¹,

Seceleanu²,

Villarreal³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

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“…According to the literature published by a large number of scholars in the past (Hartmanis and Johnson, 1982;Hemaspaandra and Williams, 2012;Bodas et al, 2021;Fang et al, 2021;Wang et al, 2022), we define 2D irregular problems as the problem in which a set of pieces that contain at least one piece of the irregular shape must be placed in a non-overlapping configuration within a given placement area in order to optimize an objective. A piece is irregular if it requires a minimum of three parameters to identify it (Bennell and Oliveira, 2008).…”

Section: Packing Problem Classificationmentioning

confidence: 99%

Two-dimensional irregular packing problems: A review

Guo

et al. 2022

Front. Mech. Eng.

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Two-dimensional (2D) irregular packing problems are widespread in manufacturing industries such as shipbuilding, metalworking, automotive production, aerospace, clothing and furniture manufacturing. Research on 2D irregular packing problems is essential for improving material utilization and industrial automation. Much research has been conducted on this problem with significant research results and certain algorithms. The work has made important contributions to solving practical problems. This paper reviews recent advances in the domain of 2D irregular packing problems based on a variety of research papers. We first introduce the basic concept and research background of 2D irregular packing problems and then summarize algorithms and strategies that have been proposed for the problems in recent years. Conclusion summarize development trends and research hotspots of typical 2D irregular shape packing problems. We hope that this review could provide guidance for researchers in the field of 2D irregular packing.

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