Exact and Parameterized Algorithms for (k, i)-Coloring

Majumdar, Diptapriyo; Neogi, Rian; Raman, Venkatesh; Tale, Prafullkumar

doi:10.1007/978-3-319-53007-9_25

Cited by 4 publications

(17 citation statements)

References 13 publications

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“…We show that (k, 1)-coloring and (k, k − 1) coloring are NP-complete, in addition to other NP-completeness results. This partially answers questions posed in [1] and [12]. -We give a 2 n n O(1) time exact algorithm for the (k, k − 1)-coloring problem.…”

Section: Introductionsupporting

confidence: 59%

“…We would like to observe a difference in the space usage of our FPT algorithm to the FPT algorithm for (q, k, i)-coloring parameterized by tree-width in [12]. We note that the algorithm in [12] can also be modified similarly to obtain an algorithm that generates a proper coloring.…”

Section: ⊓ ⊔mentioning

confidence: 99%

“…We note that the algorithm in [12] can also be modified similarly to obtain an algorithm that generates a proper coloring. However, such an algorithm would require to store all feasible colorings at each bag of the tree-decomposition, resulting in a O(( tw+1 n), which is significantly larger than the O( q k n) space required by our algorithm.…”

Section: ⊓ ⊔mentioning

confidence: 99%

“…-We give a 2 n n O(1) time exact algorithm for the (k, k − 1)-coloring problem. This is a direct improvement to the algorithm given in [12] for the same problem.…”

Section: Introductionmentioning

confidence: 97%

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On the Tractability of (k, i)-Coloring

Joshi¹,

Kalyanasundaram²,

Kare³

et al. 2018

Algorithms and Discrete Applied Mathematics

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Abstract. In an undirected graph, a proper (k, i)-coloring is an assignment of a set of k colors to each vertex such that any two adjacent vertices have at most i common colors. The (k, i)-coloring problem is to compute the minimum number of colors required for a proper (k, i)-coloring. This is a generalization of the classic graph coloring problem. Majumdar et. al. [CALDAM 2017] studied this problem and showed that the decision version of the (k, i)-coloring problem is fixed parameter tractable (FPT) with tree-width as the parameter. They asked if there exists an FPT algorithm with the size of the feedback vertex set (FVS) as the parameter without using tree-width machinery. We answer this in positive by giving a parameterized algorithm with the size of the FVS as the parameter. We also give a faster and simpler exact algorithm for (k, k − 1)-coloring, and make progress on the NP-completeness of specific cases of (k, i)-coloring.

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Section: Introductionsupporting

confidence: 59%

Section: ⊓ ⊔mentioning

confidence: 99%

Section: ⊓ ⊔mentioning

confidence: 99%

“…-We give a 2 n n O(1) time exact algorithm for the (k, k − 1)-coloring problem. This is a direct improvement to the algorithm given in [12] for the same problem.…”

Section: Introductionmentioning

confidence: 97%

See 2 more Smart Citations

On the Tractability of (k, i)-Coloring

Joshi¹,

Kalyanasundaram²,

Kare³

et al. 2018

Algorithms and Discrete Applied Mathematics

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show abstract

“…In [12], Majumdar et.al., gave an algorithm for the (q, k, i)-coloring problem in O(( q k ) tw+1 n O(1) ) time 1 , where tw denotes the tree-width of the graph. Let S be a smallest FVS of G. It is known that tw ≤ |S| + 1, see for instance [14].…”

Section: Preliminariesmentioning

confidence: 99%

On the tractability of (k,i)-coloring

Joshi

Kalyanasundaram

Kare

et al. 2021

Discrete Applied Mathematics

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In an undirected graph, a proper (k, i)-coloring is an assignment of a set of k colors to each vertex such that any two adjacent vertices have at most i common colors. The (k, i)-coloring problem is to compute the minimum number of colors required for a proper (k, i)coloring. This is a generalization of the classic graph coloring problem. Majumdar et. al. [CALDAM 2017] studied this problem and showed that the decision version of the (k, i)-coloring problem is fixed parameter tractable (FPT) with tree-width as the parameter. They asked if there exists an FPT algorithm with the size of the feedback vertex set (FVS) as the parameter without using tree-width machinery. We answer this in positive by giving a parameterized algorithm with the size of the FVS as the parameter. We also give a faster and simpler exact algorithm for (k, k − 1)-coloring, and make progress on the NP-completeness of specific cases of (k, i)-coloring. (k, i)-Coloring Problem Instance: An undirected graph G = (V, E). Output: The (k, i)-chromatic number of G, χ i k (G). We also define below the (q, k, i)-coloring problem, the decision version of the (k, i)-coloring problem. (q, k, i)-Coloring Problem Instance: An undirected graph G = (V, E). Question: Does G have a proper (k, i)-coloring using at most q colors? The (k, i)-coloring problem was first studied by Méndez-Díaz and Zabala in [1]. For arbitrary k and i, the (k, i)-coloring problem is NP-hard because (1, 0)-coloring is NP-hard. Apart from studying the basic properties, they also gave an integer linear programming formulation of the problem. Stahl [2] and independently Bollobás and Thomason [3] introduced the (k, 0)-coloring problem under the names of k-tuple coloring and k-set coloring respectively. The k-tuple coloring problem has been studied in detail [4,5], and Irving [6] showed that this problem is NP-hard as well. Some of the applications for the (k, 0)-coloring problem include construction of pseudorandom number generators, randomness extractors, secure password management schemes, aircraft scheduling, biprocessor tasks and frequency assignment to radio stations [7,8]. Brigham and Dutton [9] studied another variant of the problem, where k colors have to be assigned to each vertex such that the adjacent vertices share exactly i colors. Bonomo et. al. [10] studied the connection between the (k, i)-coloring problem on cliques and the theory of error correcting codes. In coding theory, a (j, d, k)constant weight code represents a set of codewords of length j with exactly k ones in each codeword, with Hamming distance at least d. Bonomo et. al. observed a direct connection between A(j, d, k), the largest possible size of a (j, d, k)-constant weight code, and the (k, i)-colorability of cliques and used the existing results from coding theory (such as the Johnson bound [11]) to infer results on the (k, i)-colorability of cliques. Finding bounds on A(j, d, k) is a wellstudied problem in coding theory, and lots of questions on A(j, d, k) are still open. This indicates the difficulty of the ...

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Heuristics for the Generalized Graph Coloring Problem

Kare

Kumar

2021

2021 12th International Conference on Computing Communication and Networking Technologies (ICCCNT)

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Exact and Parameterized Algorithms for (k, i)-Coloring

Cited by 4 publications

References 13 publications

On the Tractability of (k, i)-Coloring

On the Tractability of (k, i)-Coloring

On the tractability of (k,i)-coloring

Heuristics for the Generalized Graph Coloring Problem

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