Coloring half-planes and bottomless rectangles

We prove lower and upper bounds for the chromatic number of certain hypergraphs defined by geometric regions. This problem has close relations to conflict-free colorings. One of the most interesting type of regions to consider for this problem is that of the axis-parallel rectangles. We completely solve the problem for a special case of them, for bottomless rectangles. We also give an almost complete answer for half-planes and pose several open problems. Moreover we give efficient coloring algorithms

Conflict-Free Coloring Made Stronger

In FOCS 2002, Even et al. showed that any set of $n$ discs in the plane can be Conflict-Free colored with a total of at most $O(\log n)$ colors. That is, it can be colored with $O(\log n)$ colors such that for any (covered) point $p$ there is some disc whose color is distinct from all other colors of discs containing $p$. They also showed that this bound is asymptotically tight. In this paper we prove the following stronger results: \begin{enumerate} \item [(i)] Any set of $n$ discs in the plane can be colored with a total of at most $O(k \log n)$ colors such that (a) for any point $p$ that is covered by at least $k$ discs, there are at least $k$ distinct discs each of which is colored by a color distinct from all other discs containing $p$ and (b) for any point $p$ covered by at most $k$ discs, all discs covering $p$ are colored distinctively. We call such a coloring a {\em $k$-Strong Conflict-Free} coloring. We extend this result to pseudo-discs and arbitrary regions with linear union-complexity. \item [(ii)] More generally, for families of $n$ simple closed Jordan regions with union-complexity bounded by $O(n^{1+\alpha})$, we prove that there exists a $k$-Strong Conflict-Free coloring with at most $O(k n^\alpha)$ colors. \item [(iii)] We prove that any set of $n$ axis-parallel rectangles can be $k$-Strong Conflict-Free colored with at most $O(k \log^2 n)$ colors. \item [(iv)] We provide a general framework for $k$-Strong Conflict-Free coloring arbitrary hypergraphs. This framework relates the notion of $k$-Strong Conflict-Free coloring and the recently studied notion of $k$-colorful coloring. \end{enumerate} All of our proofs are constructive. That is, there exist polynomial time algorithms for computing such colorings

Making Octants Colorful and Related Covering Decomposition Problems

We give new positive results on the long-standing open problem of geometric covering decomposition for homothetic polygons. In particular, we prove that for any positive integer k, every finite set of points in R^3 can be colored with k colors so that every translate of the negative octant containing at least k^6 points contains at least one of each color. The best previously known bound was doubly exponential in k. This yields, among other corollaries, the first polynomial bound for the decomposability of multiple coverings by homothetic triangles. We also investigate related decomposition problems involving intervals appearing on a line. We prove that no algorithm can dynamically maintain a decomposition of a multiple covering by intervals under insertion of new intervals, even in a semi-online model, in which some coloring decisions can be delayed. This implies that a wide range of sweeping plane algorithms cannot guarantee any bound even for special cases of the octant problem.Comment: version after revision process; minor changes in the expositio

Dynamic Conflict-Free Colorings in the Plane

We study dynamic conflict-free colorings in the plane, where the goal is to maintain a conflict-free coloring (CF-coloring for short) under insertions and deletions. - First we consider CF-colorings of a set S of unit squares with respect to points. Our method maintains a CF-coloring that uses O(log n) colors at any time, where n is the current number of squares in S, at the cost of only O(log n) recolorings per insertion or deletion We generalize the method to rectangles whose sides have lengths in the range [1, c], where c is a fixed constant. Here the number of used colors becomes O(log^2 n). The method also extends to arbitrary rectangles whose coordinates come from a fixed universe of size N, yielding O(log^2 N log^2 n) colors. The number of recolorings for both methods stays in O(log n). - We then present a general framework to maintain a CF-coloring under insertions for sets of objects that admit a unimax coloring with a small number of colors in the static case. As an application we show how to maintain a CF-coloring with O(log^3 n) colors for disks (or other objects with linear union complexity) with respect to points at the cost of O(log n) recolorings per insertion. We extend the framework to the fully-dynamic case when the static unimax coloring admits weak deletions. As an application we show how to maintain a CF-coloring with O(sqrt(n) log^2 n) colors for points with respect to rectangles, at the cost of O(log n) recolorings per insertion and O(1) recolorings per deletion. These are the first results on fully-dynamic CF-colorings in the plane, and the first results for semi-dynamic CF-colorings for non-congruent objects

Conflict-Free Coloring of Planar Graphs

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 chi_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 chi_CF(G) <= k. For planar graphs, we obtain a tight worst-case bound: three colors are sometimes necessary and always sufficient. We also give a complete characterization of the computational complexity of conflict-free coloring. Deciding whether chi_CF(G)<= 1 is NP-complete for planar graphs G, but polynomial for outerplanar graphs. Furthermore, deciding whether chi_CF(G)<= 2 is NP-complete for planar graphs G, but always true for outerplanar graphs. 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 in {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.Comment: 30 pages, 17 figures; full version (to appear in SIAM Journal on Discrete Mathematics) of extended abstract that appears in Proceeedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2017), pp. 1951-196