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34
Interference in cellular networks: The minimum membership set cover problem
 In International Computin and Combinatoics Conference (COCOON
, 2005
"... Abstract. The infrastructure for mobile distributed tasks is often formed by cellular networks. One of the major issues in such networks is interference. In this paper we tackle interference reduction by suitable assignment of transmission power levels to base stations. This task is formalized intro ..."
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Abstract. The infrastructure for mobile distributed tasks is often formed by cellular networks. One of the major issues in such networks is interference. In this paper we tackle interference reduction by suitable assignment of transmission power levels to base stations. This task is formalized introducing the Minimum Membership Set Cover combinatorial optimization problem. On the one hand we prove that in polynomial time the optimal solution of the problem cannot be approximated more closely than with a factor ln n. On the other hand we present an algorithm exploiting linear programming relaxation techniques which asymptotically matches this lower bound. 1
Decomposition of multiple coverings into more parts
 In Proceedings of SODA
, 2009
"... We prove that for every centrally symmetric convex polygon Q, there exists a constant α such that any αkfold covering of the plane by translates of Q can be decomposed into k coverings. This improves on a quadratic upper bound proved by Pach and Tóth (SoCG’07). The question is motivated by a sensor ..."
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We prove that for every centrally symmetric convex polygon Q, there exists a constant α such that any αkfold covering of the plane by translates of Q can be decomposed into k coverings. This improves on a quadratic upper bound proved by Pach and Tóth (SoCG’07). The question is motivated by a sensor network problem, in which a region has to be monitored by sensors with limited battery life. 1
Slotted Scheduled Tag Access in MultiReader RFID Systems
"... Abstract—Radio frequency identification (RFID) is a technology where a reader device can “sense ” the presence of a closeby object by reading a tag device attached to the object. To improve coverage, multiple RFID readers can be deployed in the given region. In this paper, we consider the problem of ..."
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Abstract—Radio frequency identification (RFID) is a technology where a reader device can “sense ” the presence of a closeby object by reading a tag device attached to the object. To improve coverage, multiple RFID readers can be deployed in the given region. In this paper, we consider the problem of slotted scheduled access of RFID tags in a multiple reader environment. In particular, we develop centralized algorithms in a slotted time model to read all the tags using nearoptimal number of time slots. We consider two scenarios – one wherein the tag distribution in the physical space is unknown, and the other where tag distribution is known or can be estimated a priori. For each of these scenarios, we consider two cases depending on whether a single channel or multiple channels are available. All the above version of the problem are NPhard. We design approximation algorithms for the single channel and heuristic algorithms for the multiple channel cases. Through extensive simulations, we show that for the single channel case, our heuristics perform close to the approximation algorithms. In general, our simulations show that our algorithms significantly outperform Colorwave, an existing algorithm for similar problems. I.
State of the Union (of Geometric Objects): A Review
, 2007
"... Let C be a set of geometric objects in R d. The combinatorial complexity of the union U(C) of C is the total number of faces of all dimensions, of the arrangement of the boundaries of the objects, which lie on its boundary. We survey the known upper bounds on the complexity of the union of n geometr ..."
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Let C be a set of geometric objects in R d. The combinatorial complexity of the union U(C) of C is the total number of faces of all dimensions, of the arrangement of the boundaries of the objects, which lie on its boundary. We survey the known upper bounds on the complexity of the union of n geometric objects satisfying various natural conditions. These problems play a central role in the design and analysis of many geometric algorithms arising in robotics, molecular modeling, solid modeling, and shape matching, and the techniques used for their solutions are interesting in their own right.
Conflictfree colorings
 In Discrete and computational geometry, Algorithms Combin. 25 (Springer
"... A coloring of the elements of a planar point set P is said to be conflictfree if, for every closed disk D whose intersection with P is nonempty, there is a color that occurs in D ∩ P precisely once. We solve a problem of Even, Lotker, Ron, and Smorodinsky by showing that any conflictfree coloring ..."
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A coloring of the elements of a planar point set P is said to be conflictfree if, for every closed disk D whose intersection with P is nonempty, there is a color that occurs in D ∩ P precisely once. We solve a problem of Even, Lotker, Ron, and Smorodinsky by showing that any conflictfree coloring of every set of n points in the plane uses at least c log n colors, for an absolute constant c>0. Moreover, the same assertion is true for homothetic copies of any convex body D, in place of a disk. 1
Computing the Independence Number of Intersection Graphs
"... Computing the maximum number of disjoint elements in a collection C of geometric objects is a classical problem in computational geometry with applications ranging from frequency assignment in cellular networks to map labeling in computational cartography. The problem is equivalent to finding the in ..."
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Computing the maximum number of disjoint elements in a collection C of geometric objects is a classical problem in computational geometry with applications ranging from frequency assignment in cellular networks to map labeling in computational cartography. The problem is equivalent to finding the independence number, α(GC), of the intersection graph GC of C, obtained by connecting two elements of C with an edge if and only if their intersection is nonempty. This is known to be an NPhard task even for systems of segments in the plane with at most two different slopes. The best known polynomial time approximation algorithm for systems of arbitrary segments is due to Agarwal and Mustafa, and returns in the worst case an n 1/2+o(1)approximation for α. Using extensions of the LiptonTarjan separator theorem, we improve this result and present, for every ɛ> 0, a polynomial time algorithm for computing α(GC) with approximation ratio at most n ɛ. In contrast, for general graphs, for any ɛ> 0 it is NPhard to approximate the independence number within a factor of n 1−ɛ. We also give a subexponential time exact algorithm for computing the independence number of intersection graphs of arcwise connected sets in the plane. 1
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.
ConflictFree Coloring of Points with Respect to Rectangles and Approximation Algorithms for Discrete Independent Set
, 2012
"... In the conflictfree coloring problem, for a given range space, we want to bound the minimum value F (n) such that every set P of n points can be colored with F (n) colors with the property that every nonempty range contains a unique color. We prove a new upper bound O(n0.368) with respect to orthog ..."
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In the conflictfree coloring problem, for a given range space, we want to bound the minimum value F (n) such that every set P of n points can be colored with F (n) colors with the property that every nonempty range contains a unique color. We prove a new upper bound O(n0.368) with respect to orthogonal ranges in two dimensions (i.e., axisparallel rectangles), which is the first improvement over the previous bound O(n0.382) by Ajwani, Elbassioni, Govindarajan, and Ray [SPAA’07]. This result leads to an O(n1−0.632/2d−2) upper bound with respect to orthogonal ranges (boxes) in dimension d, and also an O(n1−0.632/(2d−3−0.368) ) upper bound with respect to dominance ranges (orthants) in dimension d ≥ 4. We also observe that combinatorial results on conflictfree coloring can be applied to the analysis of approximation algorithms for discrete versions of geometric independent set problems. Here, given a set P of (weighted) points and a set S of ranges, we want to select the largest(weight) subset Q ⊂ P with the property that every range of S contains at most one point of Q. We obtain, for example, a randomized O(n0.368)approximation algorithm for this problem with respect to orthogonal ranges in the plane. 1
Colorful Strips
 GRAPHS AND COMBINATORICS
"... We study the following geometric hypergraph coloring problem: given a planar point set and an integer k, we wish to color the points with k colors so that any axisaligned strip containing sufficiently many points contains all colors. We show that if the strip contains at least 2k−1 points, such a ..."
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We study the following geometric hypergraph coloring problem: given a planar point set and an integer k, we wish to color the points with k colors so that any axisaligned strip containing sufficiently many points contains all colors. We show that if the strip contains at least 2k−1 points, such a coloring can always be found. In dimension d, we show that the same holds provided the strip contains at least k(4lnk+ lnd) points. We also consider the dual problem of coloring a given set of axisaligned strips so that any sufficiently covered point in the plane is covered by k colors. We show that in dimension d the required coverage is at most d(k−1)+1. This complements recent impossibility results on decomposition of strip coverings with arbitrary orientations. From the computational point of view, we show that deciding whether a threedimensional point set can be 2colored so that any strip containing at least three points contains both colors is NPcomplete. This shows a big contrast with the planar case, for which this decision problem is easy.