Between coloring and list-coloring: μ-coloring

Bonomo, Flavia; Cecowski, Mariano

doi:10.1016/j.endm.2005.05.017

Cited by 10 publications

(11 citation statements)

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“…The same holds for cographs, i.e., graphs with no induced P 4 (or P 4 -free) (Hujter and Tuza 1996;Jansen and Scheffler 1997). For this class of graphs, μ-coloring is polynomial (Bonomo and Cecowski 2005). Cographs are a subclass of distance-hereditary graphs, another known subclass of perfect graphs.…”

Section: Known Resultsmentioning

confidence: 91%

“…A bipartite graph is said to be complete if its edge set includes all possible edges between V 1 and V 2 . Again, the vertex coloring problem over bipartite graphs is trivial, whereas precoloring extension (Hujter and Tuza 1993) and μ-coloring (Bonomo and Cecowski 2005) are known to be NP-complete over bipartite graphs. This implies that (γ , μ)-coloring and list-coloring over this class are also NP-complete, and that the four problems are NP-complete on comparability graphs, a widely studied subclass of perfect graphs which includes bipartite graphs.…”

Section: Known Resultsmentioning

confidence: 98%

“…Given a graph G and a function μ : V → N, G is μ-colorable if there exists a coloring f of G such that f (v) ≤ μ(v) for every v ∈ V (Bonomo and Cecowski 2005). This model arises in the context of classroom allocation to courses, where each course must be assigned a classroom which is large enough so it fits the students taking the course.…”

Section: Introductionmentioning

confidence: 98%

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Exploring the complexity boundary between coloring and list-coloring

2008

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Many classes of graphs where the vertex coloring problem is polynomially solvable are known, the most prominent being the class of perfect graphs. However, the listcoloring problem is NP-complete for many subclasses of perfect graphs. In this work we explore the complexity boundary between vertex coloring and list-coloring on such subclasses of perfect graphs where the former admits polynomial-time algorithms but the latter is NP-complete. Our goal is to analyze the computational complexity of coloring problems lying "between" (from a computational complexity viewpoint) these two problems: precoloring extension, μ-coloring, and (γ , μ)-coloring.

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Section: Known Resultsmentioning

confidence: 91%

Section: Known Resultsmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 98%

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Exploring the complexity boundary between coloring and list-coloring

2008

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“…However, the lists in the D-MSTF problem do not have an arbitrary structure, as the trains and tracks can be ordered according to length. The resulting list-coloring problem is called a -coloring problem (Bonomo and Cecowski 2005)…”

Section: Complexity Analysismentioning

confidence: 99%

Optimization Methods for Multistage Freight Train Formation

Bohlin

Gestrelius

Dahms

et al. 2016

Transportation Science

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T his paper considers mathematical optimization for the multistage train formation problem, which at the core is the allocation of classification yard formation tracks to outbound freight trains, subject to realistic constraints on train scheduling, arrival and departure timeliness, and track capacity. The problem formulation allows the temporary storage of freight cars on a dedicated mixed-usage track. This real-world practice increases the capacity of the yard, measured in the number of simultaneous trains that can be successfully handled. Two optimization models are proposed and evaluated for the multistage train formation problem. The first one is a column-based integer programming model, which is solved using branch and price. The second model is a simplified reformulation of the first model as an arc-indexed integer linear program, which has the same linear programming relaxation as the first model. Both models are adapted for rolling horizon planning and evaluated on a five-month historical data set from the largest freight yard in Scandinavia. From this data set, 784 instances of different types and lengths, spanning from two to five days, were created. In contrast to earlier approaches, all instances could be solved to optimality using the two models. In the experiments, the arc-indexed model proved optimality on average twice as fast as the column-based model for the independent instances, and three times faster for the rolling horizon instances. For the arc-indexed model, the average solution time for a reasonably sized planning horizon of three days was 16 seconds. Regardless of size, no instance took longer than eight minutes to be solved. The results indicate that optimization approaches are suitable alternatives for scheduling and track allocation at classification yards.

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“…A hierarchy of such models was studied in [4]. Two generalizations of the k-coloring problem are precoloring extension [2] and µcoloring [3].…”

Section: Introductionmentioning

confidence: 99%

On coloring problems with local constraints

Bonomo

Faenza

Oriolo

2009

Electronic Notes in Discrete Mathematics

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We deal with some generalizations of the graph coloring problem on classes of perfect graphs. Namely we consider the µ-coloring problem (upper bounds for the color on each vertex), the precoloring extension problem (a subset of vertices colored beforehand), and a problem generalizing both of them, the (γ, µ)-coloring problem (lower and upper bounds for the color on each vertex). We characterize the complexity of all those problems on clique trees of different heights, providing polytime algorithms for the cases that are easy. These results have two interesting corollaries: first, one can observe on clique trees of different heights the increasing complexity of the chain k-coloring, µ-coloring, (γ, µ)-coloring, list-coloring. Second, clique trees of height 2 are the first known example of a class of graphs where µ-coloring is polynomial time solvable and precoloring extension is NP-complete, thus being at the same time the first example where µ-coloring is polynomially solvable and (γ, µ)-coloring is NP-complete. Last, we show that the µ-coloring problem on unit interval graphs is NP-complete. These results answer three questions from [Ann. Oper. Res. 169(1) (2009), 3-16].

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Between coloring and list-coloring: μ-coloring

Cited by 10 publications

References 10 publications

Exploring the complexity boundary between coloring and list-coloring

Exploring the complexity boundary between coloring and list-coloring

Optimization Methods for Multistage Freight Train Formation

On coloring problems with local constraints

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