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ConflictFree Coloring and its Applications
, 2010
"... Let H = (V, E) be a hypergraph. A conflictfree coloring of H is an assignment of colors to V such that in each hyperedge e ∈ E there is at least one uniquelycolored vertex. This notion is an extension of the classical graph coloring. Such colorings arise in the context of frequency assignment to c ..."
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Let H = (V, E) be a hypergraph. A conflictfree coloring of H is an assignment of colors to V such that in each hyperedge e ∈ E there is at least one uniquelycolored vertex. This notion is an extension of the classical graph coloring. Such colorings arise in the context of frequency assignment to cellular antennae, in battery consumption aspects of sensor networks, in RFID protocols and several other fields, and has been the focus of many recent research papers. In this paper, we survey this notion and its combinatorial and algorithmic aspects.
Hitting and piercing rectangles induced by a point set
"... Abstract. We consider various hitting and piercing problems for the family of axisparallel rectangles induced by a point set. Selection Lemmas on induced objects are classical results in discrete geometry that have been well studied and have applications in many geometric problems like weak epsilon ..."
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Abstract. We consider various hitting and piercing problems for the family of axisparallel rectangles induced by a point set. Selection Lemmas on induced objects are classical results in discrete geometry that have been well studied and have applications in many geometric problems like weak epsilon nets and slimming Delaunay triangulations. Selection Lemma type results typically bound the maximum number of induced objects that are hit/pierced by a single point. First, we prove an exact result on the strong and the weak variant of the First Selection Lemma for rectangles. We also show bounds for the Second Selection Lemma which improve upon previous bounds when there are nearquadratic number of induced rectangles. Next, we consider the hitting set problem for induced rectangles. This is a special case of the geometric hitting set problem which has been extensively studied. We give efficient algorithms and show exact combinatorial bounds on the hitting set problem for two special classes of induced axisparallel rectangles. Finally, we show that the minimum hitting set problem for all induced lines is NPComplete.