In this paper, we consider a coverage problem for uncertain points in a tree. Let T be a tree containing a set P of n (weighted) demand points, and the location of each demand point P_i P is uncertain but is known to appear in one of m_i points on T each associated with a probability. Given a covering range , the problem is to find a minimum number of points (called centers) on T to build facilities for serving (or covering) these demand points in the sense that for each uncertain point P_i P, the expected distance from P_i to at least one center is no more than @math . The problem has not been studied before. We present an O(|T|+M ^2 M) time algorithm for the problem, where |T| is the number of vertices of T and M is the total number of locations of all uncertain points of P, i.e., M= P_i P m_i. In addition, by using this algorithm, we solve a k-center problem on T for the uncertain points of P.
If @math is a path, both the center-coverage problem and the @math -center problem on uncertain points have been studied @cite_20 , but under a somewhat special problem setting where @math is the same for all @math . The two problems were solved in @math and @math time, respectively. If @math is tree, an @math time algorithm was given in @cite_5 for the one-center problem under the above special problem setting.
@cite_5 @cite_20
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We consider a coverage problem for uncertain points in a tree. Let T be a tree containing a set ( P ) of n (weighted) demand points, and the location of each demand point (P_i P ) is uncertain but is known to be in one of (m_i ) points on T each associated with a probability. Given a covering range ( ), the problem is to find a minimum number of points (called centers) on T to build facilities for serving (or covering) these demand points in the sense that for each uncertain point (P_i P ), the expected distance from (P_i ) to at least one center is no more than ( ). The problem has not been studied before. We present an (O(|T|+M ^2 M) ) time algorithm, where |T| is the number of vertices of T and M is the total number of locations of all uncertain points of ( P ), i.e., (M= _ P_i P m_i ). We consider an interval coverage problem. Given @math intervals of the same length on a line @math and a line segment @math on @math , we want to move the intervals along @math such that every point of @math is covered by at least one interval and the sum of the moving distances of all intervals is minimized. As a basic geometry problem, it has applications in mobile sensor barrier coverage in wireless sensor networks. The previous work solved the problem in @math time. In this paper, by discovering many interesting observations and developing new algorithmic techniques, we present an @math time algorithm. We also show an @math time lower bound for this problem, which implies the optimality of our algorithm. We consider the @math -Center problem and some generalizations. For @math -Center a set of @math center vertices needs to be found in a graph @math with edge lengths, such that the distance from any vertex of @math to its nearest center is minimized. This problem naturally occurs in transportation networks, and therefore we model the inputs as graphs with bounded highway dimension, as proposed by [SODA 2010]. We show both approximation and fixed-parameter hardness results, and how to overcome them using fixed-parameter approximations, where the two paradigms are combined. In particular, we prove that for any @math computing a @math -approximation is W[2]-hard for parameter @math and NP-hard for graphs with highway dimension @math . The latter does not rule out fixed-parameter @math -approximations for the highway dimension parameter @math , but implies that such an algorithm must have at least doubly exponential running time in @math if it exists, unless the ETH fails. On the positive side, we show how to get below the approximation factor of @math by combining the parameters @math and @math : we develop a fixed-parameter @math -approximation with running time @math . Additionally we prove that, unless P=NP, our techniques cannot be used to compute fixed-parameter @math -approximations for only the parameter @math . We also provide similar fixed-parameter approximations for the weighted @math -Center and @math -Partition problems, which generalize @math -Center. The @math -center problem is a classical combinatorial optimization problem which asks to find @math centers such that the maximum distance of any input point in a set @math to its assigned center is minimized. The problem allows for elegant @math -approximations. However, the situation becomes significantly more difficult when constraints are added to the problem. We raise the question whether general methods can be derived to turn an approximation algorithm for a clustering problem with some constraints into an approximation algorithm that respects one constraint more. Our constraint of choice is privacy: Here, we are asked to only open a center when at least @math clients will be assigned to it. We show how to combine privacy with several other constraints.