Scheduling Problems and Mixed Graph Colorings

Sotskov, Yuri N.; Tanaev, V. S.; Werner, Frank

doi:10.1080/0233193021000004994

Cited by 34 publications

(34 citation statements)

References 8 publications

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“…For more on solving the multi-slot just-in-time scheduling problem via the interval graph coloring see Dereniowski and Kubiak (2010). Sotskov et al (2002), Ries and de Werra (2008) provide results on mixed graph coloring and scheduling problems.…”

Section: Lemma 4 Givenmentioning

confidence: 99%

Routing equal-size messages on a slotted ring

Dereniowski

Kubiak

2011

J Sched

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We deal with the problem of routing messages on a slotted ring network in this paper. We study the computational complexity and algorithms for this routing by means of the results known in the literature for the multi-slot justin-time scheduling problem. We consider two criteria for the routing problem: makespan, or minimum routing time, and diagonal makespan. A diagonal is simply a schedule of ring links i = 0, . . . , q − 1 in q consecutive time slots, respectively. The number of diagonals between the earliest and the latest diagonals with non-empty packets is referred to as the diagonal makespan. For the former, we show that the optimal routing of messages of size k, is NP-hard in the strong sense, while an optimal routing when k = q can be computed in O(n 2 log 2 n) time. We also give an O(n log n)-time constant factor approximation algorithm for unit size messages. For the latter, we prove that the optimal routing of messages of size k, where k divides the size of the ring q, is NP-hard in the strong sense even for any fixed k ≥ 1, while an optimal routing when k = q can be computed in O(n log n) time. We also give an O(n log n)-time approximation algorithm with an absolute error 2q − k.

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Section: Lemma 4 Givenmentioning

confidence: 99%

Routing equal-size messages on a slotted ring

Dereniowski

Kubiak

2011

J Sched

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“…For 's, , the total number of coloring methods is . Hence, the constrained-chromatic polynomial of is given by (15).…”

Section: Chromatic Polynomial Of a Digraphmentioning

confidence: 99%

“…The chromatic polynomial of a digraph or mixed graph was discussed in the literature, e.g., [14] and [15], with various definitions of coloring problems. Here, we propose an approach suitable for our application.…”

Section: Chromatic Polynomial Of a Digraphmentioning

confidence: 99%

“…To obtain the constrained-chromatic polynomial, Proposition 2 can be employed iteratively to make vertices with fixed colors the root vertices, and apply the following corollary. (15) where is the chromatic polynomial of digraph using colors, and is the descendant number of . Proof: For , the number of available colors for is .…”

Section: Chromatic Polynomial Of a Digraphmentioning

confidence: 99%

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A graphic approach to performance analysis of multistage linear interference canceller in long-code cdma systems

Hwang

Kuo

2003

IEEE Trans. Commun.

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Abstract-The signal-to-interference-plus-noise-ratio performance of the multistage linear parallel and successive interference cancellers (LPIC and LSIC) in a long-code code-division multiple-access system is analyzed with a graphic approach in this paper. The decision statistic is modeled as a Gaussian random variable, whose mean and variance can be expressed as functions of moments of R for the LPIC and L for the LSIC, respectively, where R is the correlation matrix of signature sequences and L is the strict lower triangular part of R. Since the complexity of calculating these moments increases rapidly with the growth of the stage index, a graphical representation of moments is developed to facilitate the computation. Propositions are presented to relate the moment calculation problem to several well-known problems in graph theory, i.e., the coloring, the graph decomposition, the biconnected component finding, and the Euler tour problems. It is shown that the derived analytic results match well with simulation results.Index Terms-Code-division multiple access (CDMA), linear parallel interference cancellation (LPIC), linear successive interference cancellation (LSIC), multiuser detection.

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“…Scheduling problems appearing in real-life situations may often be modeled as graph coloring problems (see [2,8,12,15,18,20]). For instance, scheduling problems involving only incompatibility constraints correspond to the classical vertex coloring problem in undirected graphs; if in addition precedence constraints occur, the problem may be handled using the vertex coloring problem in mixed graphs (i.e., graphs containing both undirected and directed edges).…”

Section: Introductionmentioning

confidence: 99%

Selective Graph Coloring in Some Special Classes of Graphs

Demange

Monnot

Pop³

et al. 2012

Lecture Notes in Computer Science

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Abstract. In this paper, we consider the selective graph coloring problem. Given an integer k ≥ 1 and a graph G = (V, E) with a partition V1, . . . , Vp of V , it consists in deciding whether there exists a set V * in G such that |V * ∩ Vi| = 1 for all i ∈ {1, . . . , p}, and such that the graph induced by V * is k-colorable. We investigate the complexity status of this problem in various classes of graphs.

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Scheduling Problems and Mixed Graph Colorings

Cited by 34 publications

References 8 publications

Routing equal-size messages on a slotted ring

Routing equal-size messages on a slotted ring

A graphic approach to performance analysis of multistage linear interference canceller in long-code cdma systems

Selective Graph Coloring in Some Special Classes of Graphs

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