4,413 research outputs found
Conflict-Free Coloring of Intersection Graphs of Geometric Objects
In FOCS'2002, Even et al. introduced and studied the notion of conflict-free
colorings of geometrically defined hypergraphs. They motivated it by frequency
assignment problems in cellular networks. This notion has been extensively
studied since then.
A conflict-free coloring of a graph is a coloring of its vertices such that
the neighborhood (pointed or closed) of each vertex contains a vertex whose
color differs from the colors of all other vertices in that neighborhood. In
this paper we study conflict-colorings of intersection graphs of geometric
objects. We show that any intersection graph of n pseudo-discs in the plane
admits a conflict-free coloring with O(\log n) colors, with respect to both
closed and pointed neighborhoods. We also show that the latter bound is
asymptotically sharp. Using our methods, we also obtain a strengthening of the
two main results of Even et al. which we believe is of independent interest. In
particular, in view of the original motivation to study such colorings, this
strengthening suggests further applications to frequency assignment in wireless
networks.
Finally, we present bounds on the number of colors needed for conflict-free
colorings of other classes of intersection graphs, including intersection
graphs of axis-parallel rectangles and of \rho-fat objects in the plane.Comment: 18 page
Triangle-free intersection graphs of line segments with large chromatic number
In the 1970s, Erdos asked whether the chromatic number of intersection graphs
of line segments in the plane is bounded by a function of their clique number.
We show the answer is no. Specifically, for each positive integer , we
construct a triangle-free family of line segments in the plane with chromatic
number greater than . Our construction disproves a conjecture of Scott that
graphs excluding induced subdivisions of any fixed graph have chromatic number
bounded by a function of their clique number.Comment: Small corrections, bibliography updat
Wireless Scheduling with Power Control
We consider the scheduling of arbitrary wireless links in the physical model
of interference to minimize the time for satisfying all requests. We study here
the combined problem of scheduling and power control, where we seek both an
assignment of power settings and a partition of the links so that each set
satisfies the signal-to-interference-plus-noise (SINR) constraints.
We give an algorithm that attains an approximation ratio of , where is the number of links and is the ratio
between the longest and the shortest link length. Under the natural assumption
that lengths are represented in binary, this gives the first approximation
ratio that is polylogarithmic in the size of the input. The algorithm has the
desirable property of using an oblivious power assignment, where the power
assigned to a sender depends only on the length of the link. We give evidence
that this dependence on is unavoidable, showing that any
reasonably-behaving oblivious power assignment results in a -approximation.
These results hold also for the (weighted) capacity problem of finding a
maximum (weighted) subset of links that can be scheduled in a single time slot.
In addition, we obtain improved approximation for a bidirectional variant of
the scheduling problem, give partial answers to questions about the utility of
graphs for modeling physical interference, and generalize the setting from the
standard 2-dimensional Euclidean plane to doubling metrics. Finally, we explore
the utility of graph models in capturing wireless interference.Comment: Revised full versio
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