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...mal points are exactly the boundary points of F ′ . Proof. First we argue that all the inward concave vertices of R must be on the outer boundary of F ′ . By the properties of geodesic shortest paths =-=[21]-=-, all the vertices of R must be vertices of F ′ . If an inward concave vertex is a vertex of the inner hole, then we can move the concave vertex outward and shrink the cycle R, as shown in Figure 9(i)...

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...ntly is NP-hard [9, 21]. Related Work. The TSP has a long and rich history of research in combinatorial optimization. It has been studied extensively in many forms, including geometric instances; see =-=[4, 13, 14, 20, 24]-=-. The problem is known to be NP-hard, even for points in the Euclidean plane [9, 21]. It has recently been shown that the geometric instances of the TSP, including the Euclidean TSP, have a polynomial...

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...ational challenge in using the UTD into realtime auralization systems is the efficient enumeration of significant early diffraction paths. Although many algorithms exist to find approximate solutions =-=[29]-=-, they are either too inefficient or prone to aliasing. For instance, Aveneau [2] enumerated all permutations of polyhedral edges within the first few Fresnel ellipsoids, which is not practical for so...

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...thms (see, e.g., [49, 13, 15]); Algorithms for dynamic versions of the problems (see, e.g., [48, 21]); Routing problems (see, e.g., [20, 57, 74]); Geometrical problems involving distances (see, e.g., =-=[28, 54, 55]-=-); and many more. Also, this short survey adopts a theoretical point of view. Problems involving distances and shortest paths in graphs are not only great mathematical problems, but are also very prac...

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...h being a pair of points in the plane [6]. Thus, we do not expect to find a PTAS for the general case. There might be a PTAS in the case of disjoint regions (the case for which we give the first constant-factor approximation); this is a fascinating open problem. For connected sets in the plane, possibly overlapping, the best known approximation ratio is O(log n), given in an algorithm discovered more than 15 years ago [14]; the running time has been improved more recently [9, 12]. For further background about the geometric TSP, the TSP with neighborhoods, and related problems, see the surveys [3, 16, 17]. 1.2 Our Contribution All prior constant-factor approximation algorithms (and PTAS’s) relied on exploiting special structure that comes from assuming that regions are either all about the same size or all are fat in some sense. Fatness was crucial in order to apply a packing argument to give effective lower bounds on the length of an optimal tour. Even the special case of horizontal line segments has had no constant-factor approximation prior to this work. Several new ideas are needed in order to address the general case in which the given regions, {P1, P2, . . . , Pn}, can be of arbitrary si...

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...exact. 2. Shortest Path Problem on a Polyhedral Surface A survey of the shortest path problem concerning a two or higher dimensional geometric object (a surface, space, network, etc.) can be found in =-=[16]-=-. We mainly discuss here about finding the shortest path between two points on a polyhedral surface. An important property of the shortest path on a polyhedron is its local optimality called unfolding...

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... P i 's. If the order in which the polygons P i must be visited is not specified, then the touring polygons problem becomes the Traveling Salesperson Problem with Neighborhoods, which is NP-hard. See =-=[20]-=-. The touring polygons problem can be modeled as a special kind of 3-dimensional shortest path problem among polyhedral obstacles. Imagine k very large sheets of paper stacked up in parallel planes or...