New polyhedral and algorithmic results on greedoids

Szeszlér, Dávid

doi:10.1007/s10107-019-01427-7

Cited by 3 publications

(11 citation statements)

References 12 publications

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“…Since both undirected and directed branching greedoids are local forest greedoids, the above theorem implies the optimality of Dijkstra's shortest path and widest path algorithms both for undirected and directed graphs (and even mixed graphs). The claim of the above theorem regarding the objective function corresponding to Example 3 played a central role in the proof of a recent result given in Szeszlér (2021) that generalized an equivalent formulation of Edmonds' classic matroid polytope theorem to local forest greedoids. The following generalization of Theorem 2 was also given in Szeszlér (2021) (with a much shorter proof than the rather technical one of Boyd (1988)).…”

Section: Further Results On the Greedoid Greedy Algorithmmentioning

confidence: 85%

“…Theorem 3 (Szeszlér 2021) Let G = (S, F ) be a local forest greedoid, P its set of paths and f : P → R a function that satisfies the following monotonicity constraints:…”

Section: Further Results On the Greedoid Greedy Algorithmmentioning

confidence: 99%

“…As it was mentioned above, (3.2) is automatically fulfilled in these cases, however, it is not at all straightforward to specialize (3.1) to order-independent objective functions. Both in Korte and Lovász (1984a) and (Korte et al 1991, Chapter XI) it is claimed that (3.1) is equivalent to the following for objective functions w : F → R: However, as it was pointed out recently in Szeszlér (2021), this reformulation clearly disregards the fact that the set α ∪ β ∪ γ , that should correspond from (3.1) to B in (3.3), need not be a feasible set. In actual fact, (3.3) does not imply (3.1), nor does it guarantee the optimality of the greedy algorithm as shown by the following trivial example: consider the undirected branching greedoid of the graph of Figure 2 with the following objective function w.…”

Section: Korte and Lovász's Theoremmentioning

confidence: 97%

“…It is claimed in (Korte et al 1991, page 156) that the greedy algorithm finds a minimum base with respect to the objective function corresponding to Example 3 in all local poset greedoids. The following simple counterexample to this claim was given in Szeszlér (2021): let S = {x, y, z, u}, F = {∅, {x}, {y}, {x, y}, {x, u}, {x, y, z}, {x, z, u}} and…”

Section: Korte and Lovász's Theoremmentioning

confidence: 99%

“…3 for the details). To the best of our knowledge, these mistakes remained hidden for more than three decades until they were pointed out and corrections were proposed in Szeszlér (2021).…”

Section: Introductionmentioning

confidence: 99%

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Sufficient conditions for the optimality of the greedy algorithm in greedoids

Szeszlér

2021

J Comb Optim

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Greedy algorithms are among the most elementary ones in theoretical computer science and understanding the conditions under which they yield an optimum solution is a widely studied problem. Greedoids were introduced by Korte and Lovász at the beginning of the 1980s as a generalization of matroids. One of the basic motivations of the notion was to extend the theoretical background behind greedy algorithms beyond the well-known results on matroids. Indeed, many well-known algorithms of a greedy nature that cannot be interpreted in a matroid-theoretical context are special cases of the greedy algorithm on greedoids. Although this algorithm turns out to be optimal in surprisingly many cases, no general theorem is known that explains this phenomenon in all these cases. Furthermore, certain claims regarding this question that were made in the original works of Korte and Lovász turned out to be false only most recently. The aim of this paper is to revisit and straighten out this question: we summarize recent progress and we also prove new results in this field. In particular, we generalize a result of Korte and Lovász and thus we obtain a sufficient condition for the optimality of the greedy algorithm that covers a much wider range of known applications than the original one.

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Section: Further Results On the Greedoid Greedy Algorithmmentioning

confidence: 85%

“…Theorem 3 (Szeszlér 2021) Let G = (S, F ) be a local forest greedoid, P its set of paths and f : P → R a function that satisfies the following monotonicity constraints:…”

Section: Further Results On the Greedoid Greedy Algorithmmentioning

confidence: 99%

Section: Korte and Lovász's Theoremmentioning

confidence: 97%

Section: Korte and Lovász's Theoremmentioning

confidence: 99%

“…3 for the details). To the best of our knowledge, these mistakes remained hidden for more than three decades until they were pointed out and corrections were proposed in Szeszlér (2021).…”

Section: Introductionmentioning

confidence: 99%

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Sufficient conditions for the optimality of the greedy algorithm in greedoids

Szeszlér

2021

J Comb Optim

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Rough set approximations based on a matroidal structure over three sets

Wang

Mao²,

Liu³

et al. 2022

Appl Intell

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Pawlak’s classical model of rough set approximations provides an efficient tool for extracting information exactly by employing available knowledge (i.e., known knowledge) in an information system, since many problems in rough set theory are NP-hard and their solution process is therefore greedy and approximate. Many extensions of Pawlak’s classical model have been proposed in recent years. Most of them are considered over one or two sets, that is, one- or two-dimensional space or one- or two-dimensional data. Aided by relation-based rough set models, a few of these extensions are considered over three sets. However, the real world is in three-dimensional space. Therefore, it is necessary to solve these problems with other models, such as covering rough set models. For this purpose, we propose the TP-matroid—a matroidal structure over three sets. Employing the family of feasible sets of a TP-matroid as the available knowledge, a pair of rough set approximations—lower and upper approximations—is provided. In addition, for an information system defined over three sets, assisted by formal concept analysis, we establish a pair of rough set approximations. Furthermore, two TP-matroids are established based on the above pair of rough set approximations. The integration between the two pairs of rough set approximations presented here is discussed. The results show that for an information system in three-dimensional space, the rough set approximations provided here can effectively explore unknown knowledge by using available knowledge based on the family of feasible sets of a TP-matroid.

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A Study on College Students’ Cognitive Intention and Path Choice of Entrepreneurship from the Perspective of Individual and Group Cooperation

Zhao

2022

Advances in Multimedia

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In order to study the overall low entrepreneurial enthusiasm of college students, a method was proposed based on the cognitive intention and path choice of college students’ entrepreneurship from the perspective of individual and group cooperation. The method briefly introduces the achievement motivation theory, entrepreneurial intention theory, entrepreneurial cognition theory, classical game theory, computer simulation, and experimental methods. Then, the research hypothesis and model construction method are elaborated, and a questionnaire is designed to study the cognitive intention and path of entrepreneurship of college students. Finally, data analysis and hypothesis testing are described to provide a guarantee for college students’ entrepreneurship. The results show that the fundamental driving force for the great progress of entrepreneurship education in colleges and universities is to meet the needs of both the collective and the individual in the top-level design and to realize the mechanism through which entrepreneurship education can achieve common development.

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New polyhedral and algorithmic results on greedoids

Cited by 3 publications

References 12 publications

Sufficient conditions for the optimality of the greedy algorithm in greedoids

Sufficient conditions for the optimality of the greedy algorithm in greedoids

Rough set approximations based on a matroidal structure over three sets

A Study on College Students’ Cognitive Intention and Path Choice of Entrepreneurship from the Perspective of Individual and Group Cooperation

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