Monday, June 8, 2009
Technische Universität Berlin
Fakultät II, Institut für Mathematik
Str. des 17. Juni 136,
room MA 041
Lecture - 14:15
In the interference scheduling problem, one is given a set of
n communication requests described by pairs of points from a metric
space. The points correspond to devices in a wireless network. In the
directed version of the problem, each pair of points consists of a
dedicated sending and a dedicated receiving device. In the
bidirectional version the devices within a pair shall be able to
exchange signals in both directions. In both versions, each pair must
be assigned a power level and a color such that the pairs in each
color class can communicate simultaneously at the specified power
levels. The feasibility of simultaneous communication within a color
class is defined in terms of the Signal to Interference Plus Noise
Ratio (SINR) that compares the strength of a signal at a receiver to
the sum of the strengths of other signals. This is commonly referred
to as the “physical model” and is the established way of
modelling interference in the engineering community. The objective is
to minimize the number of colors as this corresponds to the time
needed to schedule all requests.
We study oblivious power assignments in which the power value of a pair only depends on the distance between the points of this pair. We prove that oblivious power assignments cannot yield approximation ratios better than Omega(n) for the directed version of the problem, which is the worst possible performance guarantee as there is a straightforward algorithm that achieves an O(n)-approximation. For the bidirectional version, however, we can show the existence of a universally good oblivious power assignment: For any set of n bidirectional communication requests, the so-called square root assignment” admits a coloring with at most polylog(n) times the minimal number of colors. The proof for the existence of this coloring is non-constructive. We complement it by an approximation algorithm for the coloring problem under the square root assignment. This way, we obtain the first polynomial time algorithm with approximation ratio polylog(n) for interference scheduling in the physical model.
joint work with Alex Fanghänel, Thomas Kesselheim and Harald Räcke
Colloquium - 16:00
Since the pioneering paper of Rosenthal a lot of work has been done in order to determine classes of games that admit a potential. First, we study the existence of potential functions for weighted congestion games. Let C be an arbitrary set of locally bounded functions and let G(C) be the set of weighted congestion games with cost functions in C. We show that every weighted congestion game g in G(C) admits an exact potential if and only if C contains only affine functions. We also give a similar characterization for weighted potentials with the difference that C consists either of affine functions or of certain exponential functions. We finally extend our characterizations to weighted congestion games with facility-dependent demands and elastic demands, respectively.
First Lovasz Szekely and more recently Saharon Shelah and Alexander Soifer have presented examples of infinite geometric graphs in the plane whose chromatic numbers differ depending if we (1) allow the axiom of choice as usual or (2) demand that the colour sets be Lebesgue measurable. The existence of such graphs may be relevant to the Chromatic Number of the Plane problem. In this talk I present some more graphs with this property which are also unit distance graphs, and hence may be seen as further evidence that the chromatic number of the plane might depend on set theory. This work is an extension of work done in my bachelor's thesis.