Exact and FPT Algorithms for Max-Conflict Free Coloring in Hypergraphs

Ashok, Pradeesha; Dudeja, Aditi; Kolay, Sudeshna

doi:10.1007/978-3-662-48971-0_24

“…We present the following results; items 1-7 are for closed neighborhoods, while items [8][9][10][11] are for open neighborhoods.…”

Section: Our Contributionmentioning

confidence: 99%

“…On the combinatorial side, some authors consider the variant in which all vertices need to be colored; note that this does not change asymptotic results for general graphs and hypergraphs: it suffices to introduce one additional color for vertices that are left uncolored in our constructions. Regarding general hypergraphs, Ashok et al [8] prove that maximizing the number of conflict-freely colored edges in a hypergraph is FPT when parameterized by the number of conflict-free edges in the solution. Cheilaris et al [12] consider the case of hypergraphs induced by a set of planar Jordan regions and prove an asymptotically tight upper bound of O(log n) for the conflict-free list chromatic number of such hypergraphs.…”

Section: Related Workmentioning

confidence: 99%

See 1 more Smart Citation

Conflict-Free Coloring of Graphs

Abel¹,

Álvarez²,

Demaine³

et al. 2018

SIAM J. Discrete Math.

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A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well-studied in graph theory. Here we study the natural problem of the conflict-free chromatic number χ CF (G) (the smallest k for which conflict-free k-colorings exist). We provide results both for closed neighborhoods N [v], for which a vertex v is a member of its neighborhood, and for open neighborhoods N (v), for which vertex v is not a member of its neighborhood.For closed neighborhoods, we prove the conflict-free variant of the famous Hadwiger Conjecture: If an arbitrary graph G does not contain K k+1 as a minor, then χ CF (G) ≤ k. For planar graphs, we obtain a tight worst-case bound: three colors are sometimes necessary and always sufficient. In addition, we give a complete characterization of the algorithmic/computational complexity of conflict-free coloring. It is NP-complete to decide whether a planar graph has a conflict-free coloring with one color, while for outerplanar graphs, this can be decided in polynomial time. Furthermore, it is NP-complete to decide whether a planar graph has a conflict-free coloring with two colors, while for outerplanar graphs, two colors always suffice. For the bicriteria problem of minimizing the number of colored vertices subject to a given bound k on the number of colors, we give a full algorithmic characterization in terms of complexity and approximation for outerplanar and planar graphs.For open neighborhoods, we show that every planar bipartite graph has a conflict-free coloring with at most four colors; on the other hand, we prove that for k ∈ {1, 2, 3}, it is NP-complete to decide whether a planar bipartite graph has a conflict-free k-coloring. Moreover, we establish that any general planar graph has a conflict-free coloring with at most eight colors.2. It is NP-complete to decide whether a planar graph has a conflict-free coloring with one color.For outerplanar graphs, this question can be decided in polynomial time.

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“…Regarding general hypergraphs, Ashok et al [7] prove that maximizing the number of conflictfreely colored edges in a hypergraph is FPT when parameterized by the number of conflict-free edges in the solution. Cheilaris et al [11] consider the case of hypergraphs induced by a set of planar Jordan regions and prove an asymptotically tight upper bound of O(log n) for the conflict-free list chromatic number of such hypergraphs.…”

Section: Related Workmentioning

confidence: 99%

Three Colors Suffice: Conflict-Free Coloring of Planar Graphs

Abel¹,

Álvarez²,

Gour³

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well-studied in graph theory. Here we study the natural problem of the conflict-free chromatic number χ CF (G) (the smallest k for which conflict-free k-colorings exist), with a focus on planar graphs.For general graphs, we prove the conflict-free variant of the famous Hadwiger Conjecture: If G does not contain K k+1 as a minor, then χ CF (G) ≤ k. For planar graphs, we obtain a tight worst-case bound: three colors are sometimes necessary and always sufficient. In addition, we give a complete characterization of the algorithmic/computational complexity of conflict-free coloring. It is NP-complete to decide whether a planar graph has a conflict-free coloring with one color, while for outerplanar graphs, this can be decided in polynomial time. Furthermore, it is NP-complete to decide whether a planar graph has a conflict-free coloring with two colors, while for outerplanar graphs, two colors always suffice. For the bicriteria problem of minimizing the number of colored vertices subject to a given bound k on the number of colors, we give a full algorithmic characterization in terms of complexity and approximation for outerplanar and planar graphs.

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“…In general hypergraphs, i.e., hypergraphs not necessarily arising from a geometric context, Ashok et al [5] prove that maximizing the number of conflictfreely colored hyperedges is FPT with respect to the number of conflict-freely colored hyperedges in the solution.…”

Section: Introductionmentioning

confidence: 99%

Conflict-Free Coloring of Intersection Graphs

Fekete

¹

,

Keldenich

²

2018

Int. J. Comput. Geom. Appl.

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A conflict-free k-coloring of a graph G = (V, E) assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well studied in graph theory. Here we study the conflictfree coloring of geometric intersection graphs. We demonstrate that the intersection graph of n geometric objects without fatness properties and size restrictions may have conflict-free chromatic number in Ω(log n/ log log n) and in Ω( √ log n) for disks or squares of different sizes; it is known for general graphs that the worst case is in Θ(log 2 n). For unit-disk intersection graphs, we prove that it is NP-complete to decide the existence of a conflictfree coloring with one color; we also show that six colors always suffice, using an algorithm that colors unit disk graphs of restricted height with two colors. We conjecture that four colors are sufficient, which we prove for unit squares instead of unit disks. For interval graphs, we establish a tight worst-case bound of two.

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Exact and FPT Algorithms for Max-Conflict Free Coloring in Hypergraphs

Cited by 6 publications

References 10 publications

Conflict-Free Coloring of Graphs

Conflict-Free Coloring of Graphs

Three Colors Suffice: Conflict-Free Coloring of Planar Graphs

Conflict-Free Coloring of Intersection Graphs

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