Computational aspects of greedy partitioning of graphs

Borowiecki, Piotr

doi:10.1007/s10878-017-0185-2

Cited by 4 publications

(6 citation statements)

References 26 publications

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“…Later, Farrugia [15] has shown that if all the G i are hereditary and additive (i.e., closed under taking induced subgraphs and forming disjoint unions) then the problem is NP-hard, except the trivially polynomial case when k = 2 and both G 1 and G 2 are equal to the class of edgeless graphs. Further results in this area were obtained, e.g., by Brown [12], Alexeev et al [6], Achlioptas et al [1], or Borowiecki [10].…”

Section: Definitions and Previous Resultsmentioning

confidence: 82%

Generalized Coloring of Permutations

Jelínek,

Opler,

Valtr

2024

Algorithmica

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A permutation π is a merge of a permutation σ and a permutation τ , if we can color the elements of π red and blue so that the red elements have the same relative order as σ and the blue ones as τ . We consider, for fixed hereditary permutation classes C and D, the complexity of determining whether a given permutation π is a merge of an element of C with an element of D.We develop general algorithmic approaches for identifying polynomially tractable cases of merge recognition. Our tools include a version of nondeterministic logspace streaming recognizability of permutations, which we introduce, and a concept of bounded width decomposition, inspired by the work of Ahal and Rabinovich.As a consequence of the general results, we can provide nontrivial examples of tractable permutation merges involving commonly studied permutation classes, such as the class of layered permutations, the class of separable permutations, or the class of permutations avoiding a decreasing sequence of a given length.On the negative side, we obtain a general hardness result which implies, for example, that it is NP-complete to recognize the permutations that can be merged from two subpermutations avoiding the pattern 2413. ACM Subject ClassificationMathematics of computing → Permutations and combinations, Mathematics of computing → Combinatorial algorithms, Theory of computation → Problems, reductions and completeness

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Section: Definitions and Previous Resultsmentioning

confidence: 82%

Generalized Coloring of Permutations

Jelínek,

Opler,

Valtr

2024

Algorithmica

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“…Jensen and Toft [25]). Namely, they proved that given a graph G and a positive integer k it is NP-complete to decide if Γ(G) ≥ k. Recently, their result has been extended by Borowiecki [6] who proved that for F-free coloring an analogous problem is NP-complete for every F = K p with p ≥ 3. Despite vast literature devoted to the Grundy number, we know only few graph classes for which the greedy algorithm always outputs optimal colorings (see, e.g., Borowiecki and Rautenbach [5], and Christen and Selkow [13] for more details).…”

Section: Related Research and Our Resultsmentioning

confidence: 99%

“…On the other hand, for every fixed k ≥ 2 and connected graph F, the class of graphs F-free k-colorable with the greedy algorithm can be characterized by a finite number of minimal forbidden graphs (see, Borowiecki [6]) and hence the online variant of greedy F-free k-coloring can be solved in polynomial time. Despite the fact that the structure of minimal graphs for greedy dynamic F-free coloring seems to be more involved than in the case considered in [6], we strongly believe that Problem 1.1 can also be solved in polynomial time.…”

Section: Related Research and Our Resultsmentioning

confidence: 99%

“…In this paper we join the concept of evolving graph structure with the dynamic maintenance of temporal partitions of its vertex set into the subsets that induce graphs with specified structural properties. In addition to practical applications, significant motivation for our research on the dynamic variant of graph partitioning comes from the area of designing of graph classes along with polynomialtime approximation algorithms (see, e.g., Borowiecki [6]). In the above context, as a model for dynamic partitioning, we use the dynamic F-free coloring of graphs.…”

Section: Introductionmentioning

confidence: 99%

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Dynamic F-free Coloring of Graphs

Borowiecki

Sidorowicz

2018

Graphs and Combinatorics

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A problem of graph F-free coloring consists in partitioning the vertex set of a graph such that none of the resulting sets induces a graph containing a fixed graph F as an induced subgraph. In this paper we consider dynamic F-free coloring in which, similarly as in online coloring, the graph to be colored is not known in advance; it is gradually revealed to the coloring algorithm that has to color each vertex upon request as well as handle any vertex recoloring requests. Our main concern is the greedy approach and characterization of graph classes for which it is possible to decide in polynomial time whether for the fixed forbidden graph F and positive integer k the greedy algorithm ever uses more than k colors in dynamic F-free coloring. For various classes of graphs we give such characterizations in terms of a finite number of minimal forbidden graphs thus solving the above-mentioned problem for the so-called F-trees when F is 2-connected, and for classical trees, when F is a path of order 3 (the latter variant is also known as subcoloring or 1-improper coloring).

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“…For this purpose, let L H be any g-assignment for H . Clearly, L H | H P,R is the f | V (H P,R ) -assignment, so it has property (1) or (2). If (1) holds, then to make E not monochromatic at most one colour in L H (y) is forbidden.…”

Section: Variant 1 -Proof Of Theoremmentioning

confidence: 99%

Hypergraph operations preserving sc-greediness

Borowiecki,

Drgas-Burchardt,

Sidorowicz

2024

Discuss. Math. Graph Theory

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Given a hypergraph H and a function f : V (H) −→ N, we say that H is f -choosable if there exists a proper vertex colouring φ of H such that φThe class of sc-greedy hypergraphs is closed under the union of hypergraphs having at most one vertex in common. In this paper we consider sc-greediness of the union of hypergraphs having two vertices in common. We investigate this operation when one of the arguments is an arbitrary sc-greedy hypergraph while the second one is a hyperpath. Our research is motivated by the possibility of obtaining improved bounds on the sumchoice-number of graphs and new applications to the resource allocation problems in computer systems.

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Computational aspects of greedy partitioning of graphs

Cited by 4 publications

References 26 publications

Generalized Coloring of Permutations

Generalized Coloring of Permutations

Dynamic F-free Coloring of Graphs

Hypergraph operations preserving sc-greediness

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