Cyclic scheduling of offweekends

Koop, G. J.

doi:10.1016/0167-6377(86)90026-x

Cited by 22 publications

(5 citation statements)

References 3 publications

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“…For example, given that n = 7 and m = 4, the sequence of partitions (v 1 , v 2 , v 3 , v 4 ) = (1, 2, 1, 3) corresponds to selecting the numbers (1, 1 + 2, 1 + 2 + 1, 1 + 2 + 1 + 3) = (1, 3, 4, 7) from the numbers (1, 2, ..., 7). The selection (1,3,4,7) is cyclically equivalent to (2, 4, 5, 1), or (7,2,3,6), and so on. Alternatively, this sequence can be represented as selecting from the cyclic set (1, …, 7) the numbers (k, k + 1, k + 1 + 2, k + 1 + 2 + 1) mod 7, or (k, k + 2, k + 2 + 1, k + 2 + 1 + 3) mod 7, and so on.…”

Section: Fig (2) Examples Of Reversible and Irreversible 0-1 Maticementioning

confidence: 99%

A Dual-based Combinatorial Algorithm for Solving Cyclic Optimization Problems

Alfares

2012

CSENG

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This paper describes Patent Number U.S. 8,046,316 B2, titled "Cyclic Combinatorial Method and System", issued by the US Patents and Trademarks Office on October 25, 2011. The patent is based on a combinatorial algorithm to solve cyclic optimization problems. First, the algorithm identifies cyclically distinct solutions of such problems by enumerating cyclically distinct combinations of the basic dual variables. In combinatorial terminology, this stage of the algorithm addresses the following question: given n cyclic objects, how many cyclically distinct combinations of m (m n) objects can be selected? Integrating the operations of partition and cyclic permutation, a procedure is developed for generating cyclically distinct selections (dual solutions). Subsequently, rules are described for recognizing the set of dominant solutions. Finally, primal-dual complementary slackness relationships are used to find the primal optimum solution. This patent has many potential applications in optimization problems with cyclic 0-1 matrices, such as network problems and cyclic workforce scheduling. The patent's applicability has been illustrated by efficiently solving several cyclic labor scheduling problems.

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Section: Fig (2) Examples Of Reversible and Irreversible 0-1 Maticementioning

confidence: 99%

A Dual-based Combinatorial Algorithm for Solving Cyclic Optimization Problems

Alfares

2012

CSENG

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“…Recognition algorithms for graphs having threshold dimension 2 were first proposed by Raschle and Simon [16] and improved by Sterbini and Raschle [17]. The problem of determining the threshold dimension of a graph has several applications like aggregation of linear inequalities in integer programming [1,2], synchronization of competiting processes in a complex system such as a large computer [7,13,14,15], job scheduling [8], Guttman scales in psychology [3,9] etc.…”

Section: Chvátal and Hammermentioning

confidence: 99%

A Note on Threshold Dimension of Permutation Graphs

Bhowmick

2009

Preprint

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A graph G(V, E) is a threshold graph if there exist non-negative reals wv, v ∈ V and t such that for every U ⊆ V , P v∈U wv ≤ t if and only if U is a stable set. The threshold dimension of a graph G(V, E), denoted as t(G), is the smallest integer k such that E can be covered by k threshold spanning subgraphs of G. A permutation graph is a graph that can be represented as the intersection graph of a family of line segments that connect two parallel lines in the Euclidean plane. In this paper we will show that if G is a permutation graph then t(G) ≤ α(G) (where α(G) is the cardinality of maximum independent set in G) and this bound is tight. As a corollary we will show that t(G) ≤ n 2 where n is the number of vertices in the permutation graph G. This bound is also tight.

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“…A survey can be found in the book by Peled and Mahadev [23]. Apart from linear integer programming, threshold graphs have also been used in applications as diverse as mathematical psychology [7,8], scheduling [22] and parallel computing [24,25].…”

Section: Introductionmentioning

confidence: 99%

Certifying Fully Dynamic Algorithms for Recognition and Hamiltonicity of Threshold and Chain Graphs

et al. 2023

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Solving problems on graphs dynamically calls for algorithms to function under repeated modifications to the graph and to be more efficient than solving the problem for the whole graph from scratch after each modification. Dynamic algorithms have been considered for several graph properties, for example connectivity, shortest paths and graph recognition. In this paper we present fully dynamic algorithms for the recognition of threshold graphs and chain graphs, which are optimal in the sense that the costs per modification are linear in the number of modified edges. Furthermore, our algorithms also consider the addition and deletion of sets of vertices as well as edges. In the negative case, i.e., where the graph is not a threshold graph or chain graph anymore, our algorithms return a certificate of constant size. Additionally, we present optimal fully dynamic algorithms for the Hamiltonian cycle problem and the Hamiltonian path problem on threshold and chain graphs which return a vertex cutset as certificate for the non-existence of such a path or cycle in the negative case.

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Cyclic scheduling of offweekends

Cited by 22 publications

References 3 publications

A Dual-based Combinatorial Algorithm for Solving Cyclic Optimization Problems

A Dual-based Combinatorial Algorithm for Solving Cyclic Optimization Problems

A Note on Threshold Dimension of Permutation Graphs

Certifying Fully Dynamic Algorithms for Recognition and Hamiltonicity of Threshold and Chain Graphs

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