A cutting plane algorithm for graph coloring

Méndez-Díaz, Isabel; Zabala, Paula

doi:10.1016/j.dam.2006.07.010

Cited by 80 publications

(112 citation statements)

References 12 publications

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“…The Branch-and-Cut is based on the polyhedral study of the graph coloring polytope presented in Coll et al (2002), and it applies several families of valid inequalities, such as, for instance, the clique and the neighborhood facet defining inequalities. The same families of inequalities are used in a cutting-plane algorithm in Méndez-Díaz and Zabala (2008). An implementation of a Branchand-Price-and-Cut approach for Min-GCP is reported in Hansen et al (2009), where a family of valid inequalities that do not break the structure of the pricing subproblem is introduced.…”

mentioning

confidence: 99%

Exact Solution of Graph Coloring Problems via Constraint Programming and Column Generation

Gualandi

Malucelli

2012

INFORMS Journal on Computing

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We consider two approaches for solving the classical minimum vertex coloring problem, that is the problem of coloring the vertices of a graph so that adjacent vertices have different colors, minimizing the number of used colors, namely Constraint Programming and Column Generation. Constraint Programming is able to solve very efficiently many of the benchmarks, but suffers from the lack of effective bounding methods.On the contrary, Column Generation provides tight lower bounds by solving the fractional vertex coloring problem exploited in a Branch-and-Price algorithm, as already proposed in the literature. The Column Generation approach is here enhanced by using Constraint Programming to solve the pricing subproblem and to compute heuristic solutions. Moreover new techniques are introduced to improve the performance of the Column Generation approach in solving both the linear relaxation and the integer problem. We report extensive computational results applied to the benchmark instances: we are able to prove optimality of 11 new instances, and to improve the best known lower bounds on other 17 instances. Moreover we extend the solution approaches to a generalization of the problem known as Minimum Vertex Graph Multicoloring Problem where a given number of colors has to be assigned to each vertex.

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confidence: 99%

Exact Solution of Graph Coloring Problems via Constraint Programming and Column Generation

Gualandi

Malucelli

2012

INFORMS Journal on Computing

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“…Constraint (10) states that each guest group should be assigned to exactly one table, while (11) specifies that no pair of guest groups should be assigned to the same table if they are subject to a "Definitely Apart" constraint, with Y t = 1 when at least one guest group has been assigned to table t. Constraints (12) and (13) then ensure that a maximum of k tables are used. Finally, Constraints (14) and (15) are used to eliminate symmetries in the IP model, as suggested by Méndéz-Diaz and Zabala [12]. Without these, any particular solution using k tables could be expressed in k!…”

Section: Comparison To An Ip Modelmentioning

confidence: 99%

Creating seating plans: a practical application

Lewis

Carroll

2016

Journal of the Operational Research Society

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This paper examines the interesting problem of designing seating plans for large events such as weddings and gala dinners where, amongst other things, the aim is to construct solutions where guests are sat on the same tables as friends and family but, perhaps more importantly, are kept away from those they dislike. This problem is seen to be N P-complete from a number of different perspectives. We describe the problem model and heuristic algorithm that is used on the commercial website www.weddingseatplanner.com. We present results on the performance of this algorithm, demonstrating the factors that can influence run time and solution quality, and also present a comparison with an equivalent IP model used in conjunction with a commercial solver.

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“…For further graph-theoretical foundations, see Handbook of Graph Theory (Gross & Yellen, 2004), especially Section 5.6 . The most rigorous studies of integer programming formulations of this model, including competitive branch-and-cut implementations, are by Avella and Vasil'ev (2005) and Méndez-Díaz and Zabala (2008). For other recent research directions, see Burke and Petrovic (2002).…”

Section: An Application In Course Timetablingmentioning

confidence: 99%

“…Méndez-Díaz and Zabala (2008) compare four classes of cuts using the standard formulation and Prestwich (2003) compares five encodings of graph colouring into propositional satisfiability testing. This section elaborates on the brief overview provided in Table 1.…”

Section: Known Formulations Of Graph Colouringmentioning

confidence: 99%

“…Also, there are still dense random instances on 125 vertices from the Second DIMACS Implementation Challenge announced in 1992 (Johnson & Trick, 1996), for which the decision problem cannot be solved within reasonable time limits (Méndez-Díaz & Zabala, 2008), However, it is often possible to solve considerably larger instances in practice, by exploiting applicationspecific structure of the graphs. Springer and Thomas (1994) have, for instance, shown that graph colouring in special cases of register allocation in compilers is polynomially solvable.…”

Section: Introductionmentioning

confidence: 99%

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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 cutting plane algorithm for graph coloring

Cited by 80 publications

References 12 publications

Exact Solution of Graph Coloring Problems via Constraint Programming and Column Generation

Exact Solution of Graph Coloring Problems via Constraint Programming and Column Generation

Creating seating plans: a practical application

A supernodal formulation of vertex colouring with applications in course timetabling

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