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Some theoretical aspects of position-location problems

The position-location problem is that of computing the coordinates of a set of objects in space (usually a plane) from a sparse set of distance measurements. Because the problem is analogous to that of constructing a pin-Jointed structure from rigid bars (of given respective lengths), it is intimately linked to problems of structural rigidity. In addition to its practical significance, the problem leads to a number of surprising results and intriguing theoretical problems in geometry, combinatorics, and algorithm design. This paper presents some of the theoretical algorithmic aspects of the position-location problem; its major objective is to attract researchers to complexity problems of structural rigidity. Among the major results presented is the discovery of a large class of geometrical decision problems, all of which are randomly decidable (i.e., decidable by a probabilistic polynomial-time algorithm), but many of which seem to be Intractable.