Knapsack polytopes: a survey

Hojny, Christopher; Gally, Tristan; Habeck, Oliver; Lüthen, Hendrik; Matter, Frederic; Pfetsch, Marc E.; Schmitt, Andreas

doi:10.1007/s10479-019-03380-2

Cited by 11 publications

(5 citation statements)

References 110 publications

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“…The KP, as well as its numerous variants, has been extensively studied in the literature for over a century [25,30], with works dating as early as 1897 [32]. We refer the reader to [18,[20][21][22]25,30] for more details and the KP and its variants, and to [5,23,24] for works focused on its polyhedral aspects.…”

Section: Experiments When Optimizing Over the Cover-inequality Closurementioning

confidence: 99%

“…where each variable y j ∈ {0, 1} encodes the decision of inserting (or not) item j ∈ N into the knapsack. 1 One (or several) such constraints are present in countless real-world applications involving the solution of a MILP [23][24][25]30].…”

Section: Introductionmentioning

confidence: 99%

“…In either its original form (1) or in its lifted form (see the survey [23]), this exponentially-large family of constraints constitutes one of the earliest combinatorial inequalities adopted in a general-purpose 0-1 Integer Programming (IP) solver [17], and it it still routinely generated (heuristically) in most state-of-the-art MILP solvers. Given a (fractional) pointȳ ∈ [0, 1] n , the Separation Problem (SP) for the cover inequalities calls for an inequality of type (1) with a strictly positive cut violation, i.e., with j∈Cȳ j − |C| + 1 > 0, or for a proof that no such inequality exists.…”

Section: Introductionmentioning

confidence: 99%

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On the exact separation of cover inequalities of maximum-depth

2021

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We investigate the problem of separating cover inequalities of maximum-depth exactly. We propose a pseudopolynomial-time dynamic-programming algorithm for its solution, thanks to which we show that this problem is weakly $${\mathcal {N}}{\mathcal {P}}$$ N P -hard (similarly to the problem of separating cover inequalities of maximum violation). We carry out extensive computational experiments on instances of the knapsack and the multi-dimensional knapsack problems with and without conflict constraints. The results show that, with a cutting-plane generation method based on the maximum-depth criterion, we can optimize over the cover-inequality closure by generating a number of cuts smaller than when adopting the standard maximum-violation criterion. We also introduce the Point-to-Hyperplane Distance Knapsack Problem (PHD-KP), a problem closely related to the separation problem for maximum-depth cover inequalities, and show how the proposed dynamic programming algorithm can be adapted for effectively solving the PHD-KP as well.

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Section: Experiments When Optimizing Over the Cover-inequality Closurementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the exact separation of cover inequalities of maximum-depth

2021

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“…The 0-1 knapsack polyhedron as the most basic relaxation of a 0-1 integer program (IP) has attracted attention of many researchers over the years. In particular, developing facets for the 0-1 knapsack polyhedron has been extensively addressed over the past several decades see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] among many others. Most of the work in this direction has been focused on characterization of facets arising from lifting of the so-called minimal cover inequalities.…”

Section: Introductionmentioning

confidence: 99%

A very fast method for guaranteed generation of one facet for 0-1 knapsack polyhedron

Kianfar

Kianfar²,

Rafiee

2022

Scientia Iranica

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The 0-1 knapsack polyhedron as the most basic relaxation of a 0-1 integer program has attracted attention of many researchers over the years. We present a very fast method that is guaranteed to generate one facet for the 0-1 knapsack polyhedron. Unlike lifting of cover inequities, our method does not require an initial minimal cover or a predetermined lifting sequencing, and its worst-case complexity is linear in number of variables. Therefore, with minimal computational burden, it can be used to generate a potentially strong valid inequality based on any 0-1 relaxation of a general (mixed) integer program (M)IP. Such valid inequalities can be added to the (M)IP problem prior to solving, or given their low computational cost, can be generated during solving the (M)IP, checked to see if they separate the incumbent fractional solution, and added to the problem if they do.

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“…, n} such that j∈C a j > b. The associated cover inequality (CI) j∈C x j ≤ |C| − 1 is valid for the knapsack polytope conv(K knap ) and is not satisfied by the incidence vector of C. There is a long and rich literature on (lifted) cover inequalities for the knapsack polytope [1,12,23,10,16], and the reader is directed to the recent survey [13] for a thorough introduction.…”

Section: Introductionmentioning

confidence: 99%

Multi-cover Inequalities for Totally-Ordered Multiple Knapsack Sets: Theory and Computation

Pia¹,

Linderoth²,

Zhu³

2021

Preprint

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We propose a method to generate cutting-planes from multiple covers of knapsack constraints. The covers may come from different knapsack inequalities if the weights in the inequalities form a totally-ordered set. Thus, we introduce and study the structure of a totally-ordered multiple knapsack set. The valid multi-cover inequalities we derive for its convex hull have a number of interesting properties. First, they generalize the well-known (1, k)configuration inequalities. Second, they are not aggregation cuts. Third, they cannot be generated as a rank-1 Chvátal-Gomory cut from the inequality system consisting of the knapsack constraints and all their minimal cover inequalities. We also provide conditions under which the inequalities are facets for the convex hull of the totally-ordered knapsack set, as well as conditions for those inequalities to fully characterize its convex hull. We give an integer program to solve the separation and provide numerical experiments that showcase the strength of these new inequalities.

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Knapsack polytopes: a survey

Cited by 11 publications

References 110 publications

On the exact separation of cover inequalities of maximum-depth

On the exact separation of cover inequalities of maximum-depth

A very fast method for guaranteed generation of one facet for 0-1 knapsack polyhedron

Multi-cover Inequalities for Totally-Ordered Multiple Knapsack Sets: Theory and Computation

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