Scheduling Problems and Generalized Graph Coloring

Plurigraph Coloring and Scheduling Problems

The Electronic Journal of Combinatorics ◽

10.37236/6818 ◽

2017 ◽

Vol 24 (2) ◽

Author(s):

John Machacek

Keyword(s):

Symmetric Function ◽

Degree Sequence ◽

Simplicial Complexes ◽

Vertex Coloring ◽

Scheduling Problems ◽

Acyclic Coloring ◽

New Type ◽

Star Coloring ◽

Chromatic Symmetric Function ◽

Special Case

We define a new type of vertex coloring which generalizes vertex coloring in graphs, hypergraphs, and simplicial complexes. This coloring also generalizes oriented coloring, acyclic coloring, and star coloring. There is an associated symmetric function in noncommuting variables for which we give a deletion-contraction formula. In the case of graphs this symmetric function in noncommuting variables agrees with the chromatic symmetric function in noncommuting variables of Gebhard and Sagan. Our vertex coloring is a special case of the scheduling problems defined by Breuer and Klivans. We show how the deletion-contraction law can be applied to scheduling problems. Also, we show that the chromatic symmetric function determines the degree sequence of uniform hypertrees, but there exists pairs of 3-uniform hypertrees which are not isomorphic yet have the same chromatic symmetric function.

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On the optimality of LEPT and μc rules for parallel processors and dependent arrival processes

Advances in Applied Probability ◽

10.1017/s0001867800025854 ◽

1993 ◽

Vol 25 (04) ◽

pp. 979-996

Author(s):

Arie Hordijk ◽

Ger Koole

Keyword(s):

Dynamic Programming ◽

Parallel Processors ◽

Arrival Process ◽

Scheduling Problems ◽

Cost Structures ◽

Markov Decision ◽

New Type ◽

Arrival Processes ◽

Special Case

In this paper we study scheduling problems of multiclass customers on identical parallel processors. A new type of arrival process, called a Markov decision arrival process, is introduced. This arrival process can be controlled and allows for an indirect dependence on the numbers of customers in the queues. As a special case we show the optimality of LEPT and the µc-rule in the last node of a controlled tandem network for various cost structures. A unifying proof using dynamic programming is given.

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A categorification of the chromatic symmetric polynomial

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2527 ◽

2015 ◽

Vol DMTCS Proceedings, 27th... (Proceedings) ◽

Author(s):

Radmila Sazdanović ◽

Martha Yip

Keyword(s):

Symmetric Function ◽

Symmetric Polynomial ◽

Chromatic Polynomial ◽

Khovanov Homology ◽

Symmetric Polynomials ◽

Nous Obtenons ◽

Combinatorial Properties ◽

International Audience ◽

Chromatic Symmetric Function ◽

Frobenius Series

International audience The Stanley chromatic polynomial of a graph $G$ is a symmetric function generalization of the chromatic polynomial, and has interesting combinatorial properties. We apply the ideas of Khovanov homology to construct a homology $H$*($G$) of graded $S_n$-modules, whose graded Frobenius series $Frob_G(q,t)$ reduces to the chromatic symmetric function at $q=t=1$. We also obtain analogues of several familiar properties of the chromatic symmetric polynomials in terms of homology. Le polynôme chromatique symétrique d’un graphe $G$ est une généralisation par une fonction symétrique du polynôme chromatique, et possède des propriétés combinatoires intéressantes. Nous appliquons les techniques de l’homologie de Khovanov pour construire une homologie $H$*($G$) de modules gradués $S_n$, dont la série bigraduée de Frobeniusse $Frob_G(q,t)$ réduit au polynôme chromatique symétrique à $q=t=1$. Nous obtenons également des analogies pour plusieurs propriétés connues des polynômes chromatiques en termes d’homologie.

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Coloring Rings in Species

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2434 ◽

2014 ◽

Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽

Author(s):

Jacob White

Keyword(s):

Directed Graph ◽

Symmetric Function ◽

Chromatic Polynomial ◽

Connected Components ◽

Expansion Formula ◽

Strongly Connected Components ◽

Set Partitions ◽

Strongly Connected ◽

International Audience ◽

Chromatic Symmetric Function

International audience We present a generalization of the chromatic polynomial, and chromatic symmetric function, arising in the study of combinatorial species. These invariants are defined for modules over lattice rings in species. The primary examples are graphs and set partitions. For these new invariants, we present analogues of results regarding stable partitions, the bond lattice, the deletion-contraction recurrence, and the subset expansion formula. We also present two detailed examples, one related to enumerating subgraphs by their blocks, and a second example related to enumerating subgraphs of a directed graph by their strongly connected components.

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On the optimality of LEPT and μc rules for parallel processors and dependent arrival processes

Advances in Applied Probability ◽

10.2307/1427802 ◽

1993 ◽

Vol 25 (4) ◽

pp. 979-996 ◽

Cited By ~ 14

Author(s):

Arie Hordijk ◽

Ger Koole

Keyword(s):

Dynamic Programming ◽

Parallel Processors ◽

Arrival Process ◽

Scheduling Problems ◽

Cost Structures ◽

Markov Decision ◽

New Type ◽

Arrival Processes ◽

Special Case

In this paper we study scheduling problems of multiclass customers on identical parallel processors. A new type of arrival process, called a Markov decision arrival process, is introduced. This arrival process can be controlled and allows for an indirect dependence on the numbers of customers in the queues. As a special case we show the optimality of LEPT and the µc-rule in the last node of a controlled tandem network for various cost structures. A unifying proof using dynamic programming is given.

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A heuristic for the minimum cost chromatic partition problem

RAIRO - Operations Research ◽

10.1051/ro/2019037 ◽

2020 ◽

Vol 54 (3) ◽

pp. 845-871 ◽

Cited By ~ 1

Author(s):

Celso C. Ribeiro ◽

Philippe L. F. dos Santos

Keyword(s):

Graph Coloring ◽

Vlsi Design ◽

Minimum Cost ◽

Vertex Coloring ◽

Scheduling Problems ◽

Partition Problem ◽

Coloring Problem ◽

Graph Coloring Problem ◽

Minimum Number ◽

Trajectory Search

The graph coloring problem consists in coloring the vertices of a graph G=(V, E) with a minimum number of colors, such as that any two adjacent vertices receive different colors. The minimum cost chromatic partition problem (MCCPP) is an extension of the graph coloring problem in which there are costs associated with the colors and one seeks a vertex coloring minimizing the sum of the costs of the colors used in each vertex. The problem finds applications in VLSI design and in some scheduling problems modeled on interval graphs. We propose a trajectory search heuristic using local search, path-relinking, and perturbations for solving MCCPP and discuss computational results.

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Studying Infection Worries by Ecological Momentary Assessment: Development and Causal Factors in Sweden during the Early Phase of the Covid-19 Pandemic (Preprint)

10.2196/preprints.26743 ◽

2020 ◽

Author(s):

Peter Schulz ◽

Elin Andersson ◽

Nicole Bizzotto ◽

Margareta Norberg

Keyword(s):

Study Design ◽

Preventive Action ◽

Causal Factors ◽

Perceived Severity ◽

Assessment Development ◽

Infection Susceptibility ◽

The Face ◽

New Type ◽

Ecological Momentary ◽

Special Case

BACKGROUND The foray of Covid-19 around the globe is sure to have instigated worries in many humans, and lockdown measures may well have created their own worries. Sweden, in contrast to most other countries, had first relied on voluntary measures, but had to change its policy in the face of an increasing number of infections. OBJECTIVE The aim was to better understand the worried reactions to the virus and the lockdown measures. To grasp the reactions, their development over time was studied. METHODS Results were based on an unbalanced panel sample of 261 Swedish participants filling in 3218 interview questionnaires by smartphone in a 7-week period in 2020. Causal factors considered in this study include the perceived severity of an infection, the susceptibility of a person to the threat posed by the virus, the perceived efficacy of safeguarding measures and the assessment of government action against the spread of Covid-19. The effect of these factors on worries was traced in two analytical steps: the effects at the beginning of the study, and the effect on the trend during the study. RESULTS Findings confirmed that the hypothesized causal factors (severity of infection, susceptibility to the threat of the virus, efficacy of safeguarding and the assessment of government preventive action did indeed affect worries. CONCLUSIONS The results confirmed earlier research in a very special case and demonstrated the usefulness of a different study design, which takes a longitudinal perspective, and a new type of data analysis borrowed from multi-level study design.

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On Stanley's chromatic symmetric function and clawfree graphs

Discrete Mathematics ◽

10.1016/s0012-365x(99)00106-5 ◽

1999 ◽

Vol 205 (1-3) ◽

pp. 229-234 ◽

Cited By ~ 5

Author(s):

Vesselin Gasharov

Keyword(s):

Symmetric Function ◽

Chromatic Symmetric Function

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How to get the weak order out of a digraph ?

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2538 ◽

2015 ◽

Vol DMTCS Proceedings, 27th... (Proceedings) ◽

Author(s):

Francois Viard

Keyword(s):

Wreath Product ◽

Symmetric Function ◽

Coxeter Groups ◽

Symmetric Functions ◽

Weak Order ◽

Möbius Function ◽

Mobius Function ◽

Acyclic Digraph ◽

International Audience

International audience We construct a poset from a simple acyclic digraph together with a valuation on its vertices, and we compute the values of its Möbius function. We show that the weak order on Coxeter groups $A$$n-1$, $B$$n$, $Ã$$n$, and the flag weak order on the wreath product ℤ$r$ ≀ $S$$n$ introduced by Adin, Brenti and Roichman (2012), are special instances of our construction. We conclude by briefly explaining how to use our work to define quasi-symmetric functions, with a special emphasis on the $A$$n-1$ case, in which case we obtain the classical Stanley symmetric function. On construit une famille d’ensembles ordonnés à partir d’un graphe orienté, simple et acyclique munit d’une valuation sur ses sommets, puis on calcule les valeurs de leur fonction de Möbius respective. On montre que l’ordre faible sur les groupes de Coxeter $A$$n-1$, $B$$n$, $Ã$$n$, ainsi qu’une variante de l’ordre faible sur les produits en couronne ℤ$r$ ≀ $S$$n$ introduit par Adin, Brenti et Roichman (2012), sont des cas particuliers de cette construction. On conclura en expliquant brièvement comment ce travail peut-être utilisé pour définir des fonction quasi-symétriques, en insistant sur l’exemple de l’ordre faible sur $A$$n-1$ où l’on obtient les séries de Stanley classiques.

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

Algorithms ◽

10.3390/a14080246 ◽

2021 ◽

Vol 14 (8) ◽

pp. 246

Author(s):

Yuri N. Sotskov ◽

Еvangelina I. Mihova

Keyword(s):

Graph Coloring ◽

Precedence Constraints ◽

Scheduling Problem ◽

Scheduling Problems ◽

Due Dates ◽

Mixed Graph ◽

Coloring Problem ◽

Release Times ◽

Schedule Length ◽

Graph Coloring Problem

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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