Colorful Strips

Given a planar point set and an integer $k$, we wish to color the points with $k$ colors so that any axis-aligned strip containing enough points contains all colors. The goal is to bound the necessary size of such a strip, as a function of $k$. We show that if the strip size is at least $2k{-}1$, such a coloring can always be found. We prove that the size of the strip is also bounded in any fixed number of dimensions. In contrast to the planar case, we show that deciding whether a 3D point set can be 2-colored so that any strip containing at least three points contains both colors is NP-complete. We also consider the problem of coloring a given set of axis-aligned strips, so that any sufficiently covered point in the plane is covered by $k$ colors. We show that in $d$ dimensions the required coverage is at most $d(k{-}1)+1$. Lower bounds are given for the two problems. This complements recent impossibility results on decomposition of strip coverings with arbitrary orientations. Finally, we study a variant where strips are replaced by wedges

The Online Disjoint Set Cover Problem and its Applications

Given a universe $U$ of $n$ elements and a collection of subsets $\mathcal{S}$ of $U$, the maximum disjoint set cover problem (DSCP) is to partition $\mathcal{S}$ into as many set covers as possible, where a set cover is defined as a collection of subsets whose union is $U$. We consider the online DSCP, in which the subsets arrive one by one (possibly in an order chosen by an adversary), and must be irrevocably assigned to some partition on arrival with the objective of minimizing the competitive ratio. The competitive ratio of an online DSCP algorithm $A$ is defined as the maximum ratio of the number of disjoint set covers obtained by the optimal offline algorithm to the number of disjoint set covers obtained by $A$ across all inputs. We propose an online algorithm for solving the DSCP with competitive ratio $\ln n$. We then show a lower bound of $\Omega(\sqrt{\ln n})$ on the competitive ratio for any online DSCP algorithm. The online disjoint set cover problem has wide ranging applications in practice, including the online crowd-sourcing problem, the online coverage lifetime maximization problem in wireless sensor networks, and in online resource allocation problems.Comment: To appear in IEEE INFOCOM 201

Chromatic number of Euclidean plane

If the chromatic number of Euclidean plane is larger than four, but it is known that the chromatic number of planar graphs is equal to four, then how does one explain it? In my opinion, they are contradictory to each other. This idea leads to confirm the chromatic number of the plane about its exact value

An abstract approach to polychromatic coloring: shallow hitting sets in ABA-free hypergraphs and pseudohalfplanes

The goal of this paper is to give a new, abstract approach to cover-decomposition and polychromatic colorings using hypergraphs on ordered vertex sets. We introduce an abstract version of a framework by Smorodinsky and Yuditsky, used for polychromatic coloring halfplanes, and apply it to so-called ABA-free hypergraphs, which are a generalization of interval graphs. Using our methods, we prove that (2k-1)-uniform ABA-free hypergraphs have a polychromatic k-coloring, a problem posed by the second author. We also prove the same for hypergraphs defined on a point set by pseudohalfplanes. These results are best possible. We could only prove slightly weaker results for dual hypergraphs defined by pseudohalfplanes, and for hypergraphs defined by pseudohemispheres. We also introduce another new notion that seems to be important for investigating polychromatic colorings and epsilon-nets, shallow hitting sets. We show that all the above hypergraphs have shallow hitting sets, if their hyperedges are containment-free

Polychromatic X-ray CT Image Reconstruction and Mass-Attenuation Spectrum Estimation

We develop a method for sparse image reconstruction from polychromatic computed tomography (CT) measurements under the blind scenario where the material of the inspected object and the incident-energy spectrum are unknown. We obtain a parsimonious measurement-model parameterization by changing the integral variable from photon energy to mass attenuation, which allows us to combine the variations brought by the unknown incident spectrum and mass attenuation into a single unknown mass-attenuation spectrum function; the resulting measurement equation has the Laplace integral form. The mass-attenuation spectrum is then expanded into first order B-spline basis functions. We derive a block coordinate-descent algorithm for constrained minimization of a penalized negative log-likelihood (NLL) cost function, where penalty terms ensure nonnegativity of the spline coefficients and nonnegativity and sparsity of the density map. The image sparsity is imposed using total-variation (TV) and $\ell_1$ norms, applied to the density-map image and its discrete wavelet transform (DWT) coefficients, respectively. This algorithm alternates between Nesterov's proximal-gradient (NPG) and limited-memory Broyden-Fletcher-Goldfarb-Shanno with box constraints (L-BFGS-B) steps for updating the image and mass-attenuation spectrum parameters. To accelerate convergence of the density-map NPG step, we apply a step-size selection scheme that accounts for varying local Lipschitz constant of the NLL. We consider lognormal and Poisson noise models and establish conditions for biconvexity of the corresponding NLLs. We also prove the Kurdyka-{\L}ojasiewicz property of the objective function, which is important for establishing local convergence of the algorithm. Numerical experiments with simulated and real X-ray CT data demonstrate the performance of the proposed scheme