A Branch-And-Price Approach for Graph Multi-Coloring

Mehrotra, Anuj; Trick, Michael A.

doi:10.1007/978-0-387-48793-9_2

Cited by 17 publications

(8 citation statements)

References 16 publications

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“…Malaguti and Toth [56] use a combination of tabu search and population management procedures as a metaheuristic to solve (slightly generalized) multicoloring problems. Mehrotra and Trick [60] apply branch and price to generate independent sets for solving the multicoloring problem.…”

Section: ϑ K (G) and χ K (G) Formulations And Their Relationmentioning

confidence: 99%

On the generalized $\vartheta$-number and related problems for highly symmetric graphs

Sinjorgo¹,

Sotirov²

2021

Preprint

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This paper is an in-depth analysis of the generalized ϑ-number of a graph. The generalized ϑ-number, ϑ k (G), serves as a bound for both the k-multichromatic number of a graph and the maximum k-colorable subgraph problem. We present various properties of ϑ k (G), such as that the series (ϑ k (G)) k is increasing and bounded above by the order of the graph G. We study ϑ k (G) when G is the graph strong, disjunction and Cartesian product of two graphs. We provide closed form expressions for the generalized ϑ-number on several classes of graphs including the Kneser graphs, cycle graphs, strongly regular graphs and orthogonality graphs. Our paper provides bounds on the product and sum of the k-multichromatic number of a graph and its complement graph, as well as lower bounds for the k-multichromatic number on several graph classes including the Hamming and Johnson graphs.

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Section: ϑ K (G) and χ K (G) Formulations And Their Relationmentioning

confidence: 99%

On the generalized $\vartheta$-number and related problems for highly symmetric graphs

Sinjorgo¹,

Sotirov²

2021

Preprint

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“…When the pricing problem is dicult to solve, heuristic and local search methods are generally used in order to nd columns with negative reduced costs (Mehrotra and Trick, 2007). The subproblem is then only solved to completion in order to prove that an optimal solution to LRMP is also optimal o LMP.…”

Section: A Class-based Sparse Formulationmentioning

confidence: 99%

Scheduling policies for multi-period services

Nunez-del-Toro

Fernández

Kalcsics

et al. 2016

European Journal of Operational Research

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This paper discusses a multi-period service scheduling problem. In this problem, a set of customers is given who periodically require service over a finite time horizon. To satisfy the service demands, a set of operators is given, each with a fixed capacity in terms of the number of customers an operator can serve per period. The task is to determine for each customer the periods in which he will be visited by an operator such that the periodic service requests of the customers are adhered to and the total number of operators used over the time horizon is minimal. Two alternative policies for scheduling customer visits are considered. In the first one, a customer is visited just on time, i.e., in the period where he or she has a demand for service. The second policy allows service visits ahead of time. The rationale behind this policy is that allowing irregular visits may reduce the overall number of operators needed throughout the time horizon. To solve the problem, integer linear programming formulations are proposed for both policies and numerical experiments are presented that show the reduction in the number of operators used when visits ahead of time are allowed. As only small instances can be solved optimally, a heuristic algorithm is introduced in order to obtain good quality solutions and shorter computing times.Peer ReviewedPostprint (author's final draft

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“…Other variants of the problems are surveyed by Halldórsson and Kortsarz (2004) and Aardal et al (2007). Mehrotra and Trick (2007) seem to have the present-best solver for multicolouring.…”

Section: Lemma 3 Algorithm a Runs In Time O(|v | |E|)mentioning

confidence: 99%

A supernodal formulation of vertex colouring with applications in course timetabling

et al. 2010

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For many problems in Scheduling and Timetabling the choice of an mathematical programming formulation is determined by the formulation of the graph colouring component. This paper briefly surveys seven known integer programming formulations of vertex colouring and introduces a new formulation using "supernodes". In the definition of George and McIntyre [SIAM J. Numer. Anal. 15 (1978), no. 1, 90-112], "supernode" is a complete subgraph, where each two vertices have the same neighbourhood outside of the subgraph. Seen another way, the algorithm for obtaining the best possible partition of an arbitrary graph into supernodes, which we give and show to be polynomial-time, makes it possible to use any formulation of vertex multicolouring to encode vertex colouring. The power of this approach is shown on the benchmark problem of Udine Course Timetabling. Results from empirical tests on DIMACS colouring instances, in addition to instances from other timetabling applications, are also provided and discussed.

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A Branch-And-Price Approach for Graph Multi-Coloring

Cited by 17 publications

References 16 publications

On the generalized $\vartheta$-number and related problems for highly symmetric graphs

On the generalized $\vartheta$-number and related problems for highly symmetric graphs

Scheduling policies for multi-period services

A supernodal formulation of vertex colouring with applications in course timetabling

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