Mixed graph colouring for unit-time scheduling

Kouider, Ahmed; Haddadene, Hacene Ait; Ourari, Samia; Oulamara, Ammar

doi:10.1080/00207543.2016.1224950

Cited by 11 publications

(10 citation statements)

References 17 publications

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“…In article [7], it is shown that any job-shop scheduling problem with unit processing times of the operations to minimize the total completion time ∑ C i may be represented as a mixed graph coloring with the minimization of the sum of colors of path-endpoints in the mixed graph G = (V, A, E); see [8,9]. Hereafter, C i denotes a completion time of the job J i .…”

Section: Theorem 1 ([3])mentioning

confidence: 99%

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Scheduling Multiprocessor Tasks with Equal Processing Times as a Mixed Graph Coloring Problem

Sotskov

Mihova²

2021

Algorithms

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This article extends the scheduling problem with dedicated processors, unit-time tasks, and minimizing maximal lateness for integer due dates to the scheduling problem, where along with precedence constraints given on the set of the multiprocessor tasks, a subset of tasks must be processed simultaneously. Contrary to a classical shop-scheduling problem, several processors must fulfill a multiprocessor task. Furthermore, two types of the precedence constraints may be given on the task set . We prove that the extended scheduling problem with integer release times of the jobs to minimize schedule length may be solved as an optimal mixed graph coloring problem that consists of the assignment of a minimal number of colors (positive integers) to the vertices of the mixed graph such that, if two vertices and are joined by the edge , their colors have to be different. Further, if two vertices and are joined by the arc , the color of vertex has to be no greater than the color of vertex . We prove two theorems, which imply that most analytical results proved so far for optimal colorings of the mixed graphs , have analogous results, which are valid for the extended scheduling problems to minimize the schedule length or maximal lateness, and vice versa.

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Section: Theorem 1 ([3])mentioning

confidence: 99%

“…In articles [4,5,8,9], it is shown that the job-shop scheduling problem J|t i = 1|C max may be represented as an optimal coloring c(G) of the specified mixed graph G = (V, A, E). Based on Definition 1, one can represent every job J k ∈ J as a union of the path…”

Section: A Unit-time Minimum-length Job-shop Scheduling Problemmentioning

confidence: 99%

Scheduling Multiprocessor Tasks with Equal Processing Times as a Mixed Graph Coloring Problem

Sotskov

Mihova²

2021

Algorithms

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“…Algorithms for constructing an optimal c < -coloring of the mixed graph G = (V, A, E) have been derived in [39,40]. In [40], it is shown that an optimal c < -coloring may be constructed for the mixed graph…”

Section: Theorem 24mentioning

confidence: 99%

“…In [39], the scheduling problem J|p i = 1|C max is presented as finding an optimal c < -coloring of a special mixed graph satisfying Properties 1 and 2. The lower and upper bounds on the strict chromatic number χ < (G) have been proven.…”

Section: K R Kmentioning

confidence: 99%

Mixed Graph Colorings: A Historical Review

Sotskov

2020

Mathematics

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This paper presents a historical review and recent developments in mixed graph colorings in the light of scheduling problems with the makespan criterion. A mixed graph contains both a set of arcs and a set of edges. Two types of colorings of the vertices of the mixed graph and one coloring of the arcs and edges of the mixed graph have been considered in the literature. The unit-time scheduling problem with the makespan criterion may be interpreted as an optimal coloring of the vertices of a mixed graph, where the number of used colors is minimum. Complexity results for optimal colorings of the mixed graph are systematized. The published algorithms for finding optimal mixed graph colorings are briefly surveyed. Two new colorings of a mixed graph are introduced.

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“…The latter paper provides some general bounds on the chromatic number and an O(n 2 ) time algorithm for trees. Further work has been devoted to heuristics for general mixed graphs and their applications, fast algorithms on special graph classes, as well as several hardness results (Al-Anzi et al 2006;Furmanczyk et al 2007;Gendron et al 2007;Kouider et al 2017;Ries and de Werra 2008;Sotskov et al 2001). To mention one example, Mixed Graph Coloring restricted to bipartite graphs is NP-complete even with 3 colors and further restrictions, and it is solvable in polynomial time when only 2 colors are permitted (Ries 2007(Ries , 2010.…”

Section: Related Literature and Complexity Statusmentioning

confidence: 99%

Parameterized Mixed Graph Coloring

Damaschke

2019

J Comb Optim

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Coloring of mixed graphs that contain both directed arcs and undirected edges is relevant for scheduling of unit-length jobs with precedence constraints and conflicts. The classic GHRV theorem (attributed to Gallai, Hasse, Roy, and Vitaver) relates graph coloring to longest paths. It can be extended to mixed graphs. In the present paper we further extend the GHRV theorem to weighted mixed graphs. As a byproduct this yields a kernel and a parameterized algorithm (with the number of undirected edges as parameter) that is slightly faster than the brute-force algorithm. The parameter is natural since the directed version is polynomial whereas the undirected version is NPcomplete. Furthermore we point out a new polynomial case where the edges form a clique.

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Mixed graph colouring for unit-time scheduling

Cited by 11 publications

References 17 publications

Scheduling Multiprocessor Tasks with Equal Processing Times as a Mixed Graph Coloring Problem

Scheduling Multiprocessor Tasks with Equal Processing Times as a Mixed Graph Coloring Problem

Mixed Graph Colorings: A Historical Review

Parameterized Mixed Graph Coloring

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