The Vertex Coloring Problem and its generalizations

Malaguti, Enrico

doi:10.1007/s10288-008-0071-y

Cited by 19 publications

(13 citation statements)

References 77 publications

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“…In case of obtaining a maximal set s as the outcome for more than one s ∈ S, set the value of the corresponding variable as the sum of the original ones. Finally, constraints (11) restrict the number of colors assigned to pairs of adjacent vertices in G and constraints (12) require variables x s , s ∈ S, to be nonnegative integers. Unless otherwise stated, through the rest of the article we assume S ⊆ 2 V including maximal sets only.…”

Section: Modelsmentioning

confidence: 99%

“…Throughout the article, we assume that k > c and that

| R | \geq χ_{k}^{c} (G)

. The problem is

N P

‐Hard in the general case and reduces to the VCP when k = 1 and c = 0 (see Garey and Johnson for complexity results on VCP, and Malaguti and Malaguti and Toth for other

N P

‐Hard generalizations of the VCP). The problem remains

N P

‐Hard for the special case when

c = k - 1

(see Méndez‐Díaz and Zabala ) while the complexity when

c < k - 1, c > 0

, is open.…”

Section: Introductionmentioning

confidence: 99%

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A branch‐and‐price algorithm for the (k,c)‐coloring problem

et al. 2014

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In this article, we study the (k,c)-coloring problem, a generalization of the vertex coloring problem where we have to assign k colors to each vertex of an undirected graph, and two adjacent vertices can share at most c colors. We propose a new formulation for the (k,c)-coloring problem and develop a Branch-and-Price algorithm. We tested the algorithm on instances having from 20 to 80 vertices and different combinations for k and c, and compare it with a recent algorithm proposed in the literature. Computational results show that the overall approach is effective and has very good performance on instances where the previous algorithm fails.

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Section: Modelsmentioning

confidence: 99%

“…Throughout the article, we assume that k > c and that

| R | \geq χ_{k}^{c} (G)

. The problem is

N P

‐Hard in the general case and reduces to the VCP when k = 1 and c = 0 (see Garey and Johnson for complexity results on VCP, and Malaguti and Malaguti and Toth for other

N P

‐Hard generalizations of the VCP). The problem remains

N P

‐Hard for the special case when

c = k - 1

(see Méndez‐Díaz and Zabala ) while the complexity when

c < k - 1, c > 0

, is open.…”

Section: Introductionmentioning

confidence: 99%

A branch‐and‐price algorithm for the (k,c)‐coloring problem

et al. 2014

Self Cite

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“…To avoid the destruction of the pricing problem structure during column generation, they propose a branching scheme that modifies the structure of the graph by adding edges or merging vertices. Malaguti (2008) describes a new algorithm for graph coloring that combines the branch-andprice algorithm with an effective initialization heuristic. This heuristic method employs local search techniques together with an evolutionary algorithm to generate an initial solution as well as a good initial pool to seed the column generation procedure (Malaguti et al, 2008).…”

Section: Introductionmentioning

confidence: 99%

A Wide Branching Strategy for the Graph Coloring Problem

Morrison

Sauppe

Sewell

et al. 2014

INFORMS Journal on Computing

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Branch-and-price algorithms for the graph coloring problem use an exponentially-sized independent set based integer programming formulation to produce usually tight lower bounds to enable more aggressive pruning in the branch-and-bound tree. One major problem inherent to any branch-and-price scheme for graph coloring is that in order to avoid destroying the pricing problem structure during column generation, difficult-to-implement branching rules that modify the underlying graph must be used. This paper proposes an alternate branching strategy that does not change the graph in order to solve the pricing problem, but rather modifies the search tree to require fewer calls to difficult instances of the pricing problem. This approach, called wide branching, generates many subproblems at each node in the branch-and-price tree, which significantly reduces the length of any path through the search tree. In contrast, traditional deep branching only creates two subproblems per node, assigning a variable to either zero or one. A delayed branching procedure is introduced that prevents the branching factor at any particular node from growing too large in this scheme. Finally, computational results are presented that show the wide branching strategy to be competitive with state-of-the-art graph coloring solvers.

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“…Every node has a weight, which corresponds to the size of the channel. Minimizing buffer memory consumption, therefore, reduces to the NP-hard weighted vertex coloring problem [13], [14]: a graph G is colored with p colors such that no two adjacent vertices are of the same color. We denote the maximum weight of a vertex colored with color i as max(i), and we need to find a coloring such that ∑ p i=1 max(i) is minimum.…”

Section: Tackling State Space Explosionmentioning

confidence: 99%

Buffer Sharing in Rendezvous Programs

Vasudevan

Edwards

2010

IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.

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Abstract-Most compilers focus on optimizing performance, often at the expense of memory, but efficient memory use can be just as important in constrained environments such as embedded systems. This paper presents a memory reduction technique for CSP-style rendezvous communication, which is applied to the deterministic concurrent programming language SHIM. It focuses on reducing memory consumption by sharing communication buffers among tasks. It determines pairs of buffers that can never be in use simultaneously and use a shared region of memory for each pair. The technique produces a static abstraction of a SHIM program's dynamic behavior, which is then analyzed to find buffers that are never occupied simultaneously. Experiments show the technique runs quickly on modest-sized programs and can sometimes reduce memory requirements by half.

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The Vertex Coloring Problem and its generalizations

Cited by 19 publications

References 77 publications

A branch‐and‐price algorithm for the (k,c)‐coloring problem

A branch‐and‐price algorithm for the (k,c)‐coloring problem

A Wide Branching Strategy for the Graph Coloring Problem

Buffer Sharing in Rendezvous Programs

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