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Three Colors Suffice: Conflict-Free Coloring of Planar Graphs

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Three Colors Suffice: Conflict-Free Coloring of Planar Graphs Three Colors Suffice: Conflict-Free Coloring of Planar Graphs The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Abel, Zachary, et al. "Three Colors Suffice: Conflict-Free Coloring of Planar Graphs." Proceedings of the Twenty-Eighth Annual ACM- SIAM Symposium on Discrete Algorithms, 16-19 January, 2017, Barcelona, Spain, Society for Industrial and Applied Mathematics, 2017, pp. 1951–63. As Published http://dx.doi.org/10.1137/1.9781611974782.127 Version Author's final manuscript Citable link http://hdl.handle.net/1721.1/114769 Terms of Use Creative Commons Attribution-Noncommercial-Share Alike Detailed Terms http://creativecommons.org/licenses/by-nc-sa/4.0/ Three Colors Suffice: Conflict-Free Coloring of Planar Graphs Zachary Abel∗ Victor Alvarezy Erik D. Demainez Sándor P. Feketey Aman Gourx Adam Hesterberg∗ Phillip Keldenichy Christian Scheffery Abstract a planar graph [5, 6, 26] was a tantalizing open problem A conflict-free k-coloring of a graph assigns one of k for more than 100 years; the quest for solving this chal- different colors to some of the vertices such that, for lenge contributed to the development of graph theory, every vertex v, there is a color that is assigned to exactly but also to computers in theorem proving [29]. A gener- one vertex among v and v’s neighbors. Such colorings alization that is still unsolved is the Hadwiger Conjec- have applications in wireless networking, robotics, and ture [18]: A graph is k-colorable if it has no Kk+1 minor. geometry, and are well-studied in graph theory. Here we Over the years, there have been many variations study the natural problem of the conflict-free chromatic on coloring, often motivated by particular applications. One such context is wireless communication, where number χCF (G) (the smallest k for which conflict-free k-colorings exist), with a focus on planar graphs. “colors” correspond to different frequencies. This also For general graphs, we prove the conflict-free vari- plays a role in robot navigation, where different beacons ant of the famous Hadwiger Conjecture: If G does not are used for providing direction. To this end, it is vital that in any given location, a robot is adjacent to a contain Kk+1 as a minor, then χCF (G) k. For planar graphs, we obtain a tight worst-case bound:≤ three colors beacon with a frequency that is unique among the ones are sometimes necessary and always sufficient. In addi- that can be received. This notion has been introduced tion, we give a complete characterization of the algorith- as conflict-free coloring, formalized as follows. For any vertex v V of a simple graph G = (V; E), the closed mic/computational complexity of conflict-free coloring. 2 It is NP-complete to decide whether a planar graph has neighborhood N[v] consists of all vertices adjacent to v a conflict-free coloring with one color, while for outer- and v itself. A conflict-free k-coloring of G assigns one of k different colors to a (possibly proper) subset S V planar graphs, this can be decided in polynomial time. ⊆ Furthermore, it is NP-complete to decide whether a pla- of vertices, such that for every vertex v V , there is a vertex y N[v], called the conflict-free2 neighbor nar graph has a conflict-free coloring with two colors, 2 while for outerplanar graphs, two colors always suffice. of v, such that the color of y is unique in the closed For the bicriteria problem of minimizing the number of neighborhood of v. The conflict-free chromatic number colored vertices subject to a given bound k on the num- χCF (G) of G is the smallest k for which a conflict-free ber of colors, we give a full algorithmic characterization coloring exists. Observe that χCF (G) is bounded from in terms of complexity and approximation for outerpla- above by the proper chromatic number χ(G) because nar and planar graphs. in a proper coloring, every vertex is its own conflict- arXiv:1701.05999v1 [cs.DM] 21 Jan 2017 free neighbor. 1 Introduction Conflict-free coloring has received an increasing amount of attention. Because of the relationship to Coloring the vertices of a graph is one of the fundamen- classic coloring, it is natural to investigate the conflict- tal problems in graph theory, both scientifically and his- free coloring of planar graphs. In addition, previous torically. Proving that four colors always suffice to color work has considered either general graphs and hyper- graphs (e.g., see [25]) or geometric scenarios (e.g., see ∗Mathematics Department, MIT, Cambridge, Mas- [20]); we give a more detailed overview further down. sachusetts, USA This adds to the relevance of conflict-free coloring of yAlgorithms Group, TU Braunschweig, Braun- planar graphs, which constitute the intersection of gen- schweig, Germany eral graphs and geometry. In addition, the subclass of zComputer Science and Artificial Intelligence Laboratory, MIT, Massachusetts, USA outerplanar graphs is of interest, as it corresponds to xComputer Science and Engineering, IIT Bombay, Mum- subdividing simple polygons by chords. bai, India There is a spectrum of different scientific challenges avoid interference, there must be at least one base sta- when studying conflict-free coloring. What are worst- tion in range using a unique frequency for every point case bounds on the necessary number of colors? When in the entire covered area. The task is to assign a fre- is it NP-hard to determine the existence of a conflict- quency to each base station minimizing the number of free k-coloring, when polynomially solvable? What frequencies. On an abstract level, this induces a color- can be said about approximation? Are there sufficient ing problem on a hypergraph where the base stations conditions for more general graphs? And what can be correspond to the vertices and there is an hyperedge said about the bicriteria problem, in which also the between some vertices if the range of the corresponding number of colored vertices is considered? We provide base stations has a non-empty common intersection. extensive answers for all of these aspects, basically If the hypergraph is induced by disks, Even et providing a complete characterization for planar and al. [15] prove that (log n) colors are always sufficient. outerplanar graphs. Alon and SmorodinskyO [4] extend this by showing that each family of disks, where each disk intersects at most 1.1 Our contribution. We present the following k others, can be colored using (log3 k) colors. Further- results. more, for unit disks, Lev-TovO and Peleg [23] present an (1)-approximation algorithm for the number of colors. 1. For general graphs, we provide the conflict-free HorevO et al. [21] extend this by showing that any set of variant of the Hadwiger Conjecture: If G does not n disks can be colored with (k log n) colors, even if O contain Kk+1 as a minor, then χCF (G) k. every point must see k distinct unique colors. Abam et ≤ al. [1] discuss the problem in the context of cellular net- 2. It is NP-complete to decide whether a planar graph works where the network has to be reliable even if some has a conflict-free coloring with one color. For number of base stations fault, giving worst-case bounds outerplanar graphs, this question can be decided for the number of colors required. in polynomial time. For the dual problem of coloring a set of points such that each region from some family of regions con- 3. It is NP-complete to decide whether a planar graph tains at least one uniquely colored point, Har-Peled and has a conflict-free coloring with two colors. For Smorodinsky [19] prove that with respect to every fam- outerplanar graphs, two colors always suffice. ily of pseudo-disks, every set of points can be colored 4. Three colors are sometimes necessary and always using (log n) colors. For rectangle ranges, Elbassioni O sufficient for conflict-free coloring of a planar graph. and Mustafa [14] show that it is possible to add a sub- linear number of points such that a conflict-free color- 3=8 (1+") 5. For the bicriteria problem of minimizing the num- ing with (n · ) colors becomes possible. Ajwani ber of colored vertices subject to a given bound et al. [2]O complement this by showing that coloring a χ (G) k with k 1; 2 , we prove that the set of points with respect to rectangle ranges is always CF ≤ 2 f g problem is NP-hard for planar and polynomially possible using (n0:382) colors. For coloring points on solvable in outerplanar graphs. a line with respectO to intervals, Cheilaris et al. [10] 2 present a 2-approximation algorithm, and a 5 k - 6. For planar graphs and k = 3 colors, minimizing approximation algorithm when every interval must− see the number of colored vertices does not have a k uniquely colored vertices. Hoffman et al. [20] give constant-factor approximation, unless P = NP. tight bounds for the conflict-free chromatic art gallery problem under rectangular visibility in orthogonal poly- 7. For planar graphs and k 4 colors, it is NP- gons: Θ(log log n) are sometimes necessary and always complete to minimize the number≥ of colored ver- sufficient. Chen et al. [13] consider the online version tices. The problem is fixed-parameter tractable of the conflict-free coloring of a set of points on the (FPT) and allows a PTAS. line, where each newly inserted point must be assigned a color upon insertion, and at all times the coloring has 1.2 Related work.
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