Survey on Decomposition of Multiple Coverings

Abstract

)>IJH=?J The study of multiple coverings was initiated by Davenport and L. Fejes Tóth more than 50 years ago. In 1980 and 1986, the rst named author published the rst papers about decomposability of multiple coverings. It was discovered much later that, besides its theoretical interest, this area has practical applications to sensor networks. Now there is a lot of activity in this eld with several breakthrough results, although, many basic questions are still unsolved. In this survey, we outline the most important results, methods, and questions.1 Cover-decomposability and the sensor cover problem De nition 1.1. A planar set P is said to be cover-decomposable if there exists a (minimal) constant m = m(P ) such that every m-fold covering of the plane with translates of P can be decomposed into two coverings.Note that the above term is slightly misleading: we decompose (partition) not the set P , but a collection P of its translates. Such a partition is sometimes regarded a coloring of the members of P.

Octants Are Cover-Decomposable

“…As containment is preserved, coloring the positive octants with apices of the original point set according to Theorem 1, the same coloring for the vertices gives a valid coloring for Theorem 6. The reverse implication is similar; it again uses that containment is preserved by this dualization (for more on dualization see the surveys [14] and [11]). …”

“…For more about handling these issues and other results on cover-decomposability, see the recent surveys [14] and [11] and the papers [2,3,5,9,10,13,[15][16][17].…”

“…We call such a two-coloring of a point set in the space a good coloring. If we can do such a coloring for any P , then it follows using a standard dualization argument (see [14] or [11]) that W (and thus any octant) is cover-decomposable. So from now on our goal will be to show the existence of such a coloring.…”

An Abstract Approach to Polychromatic Coloring: Shallow Hitting Sets in ABA-free Hypergraphs and Pseudohalfplanes

“…The primal and the dual versions are equivalent if the underlying regions are the translates of some fixed set. For the proof of this statement and an extensive survey of results related to cover-decomposition, see e.g., [19]. Below we mention some of these results, stated in the equivalent primal form.…”

“…Moreover, the coloring of geometric shapes in the plane is related to the problems of cover-decomposability, conflict-free colorings and ǫ-nets; these problems have applications in sensor networks and frequency assignment as well as other areas. For surveys on these and related problems see [19,26].…”