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The paper studies the problem of computing a minimal energy cost range assignment in an ad-hoc wireless network which allows a station s to perform a broadcast operation in at most h hops. The general version of the problem (i.e. when transmission costs are arbitrary) is known to be log-APX hard even for h=2. The current paper considers the well-studied real case in which n stations are located in the plane and the cost to transmit from station i to station j is proporttional to the a-th power of the distance between i and j, where a is any positive constant. A polynomial-time algorithm is presented for finding an optimal range assignment to perform a 2-hop broadcast from a given source station. The algorithm relies on dynamic programming and operates in (worst case) O(n^7) time. Then, a polynomial-time approximation scheme (PTAS) is provided for the above problem for any fixed h>=1

a b s t r a c tThe paper considers the geometric conflict-free coloring problem, introduced in [G. Even, Z. Lotker, D. Ron, S. Smorodinsky, Conflict-free colorings of simple geometric regions with applications to frequency assignment in cellular networks, SIAM J. Comput. 33 (2003) 94-133]. The input of this problem is a set of regions in the plane and the output is an assignment of colors to the regions, such that for every point p in the total coverage area, there exist a color i and a region D such that p ∈ D, the region D is colored i, and every other region D that contains p is not colored i. The target is to minimize the number of colors used. This problem arises from issues of frequency assignment in radio networks. The paper presents an O(1) approximation algorithm for the conflict-free coloring problem where the regions are unit disks.

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