Computational Geometry Column 42

A compendium of thirty previously published open problems in computational geometry is presented. [SIGACT News, 32(3) Issue, 120 Sep. 2001, 63–72.] The computational geometry community has made many advances in the relatively short (quarter-century) of the field’s existence. Along the way researchers have engaged with a number of problems that have resisted solution. We gather here a list of open problems in computational geometry (and closely related disciplines) which together have occupied a sizable portion of the community’s efforts over the last decade or more. We make no claim to comprehensiveness, only that were all these problems to be solved, the field would be greatly advanced. All the problems have appeared in earlier publications, but we believe all remain open as stated. We present them in condensed form, without always defining every technical term, but in each case providing at least one reference for further investigation. Our list consists of predominantly theoretical questions for which the problem can be succinctly stated and the measure of success is clear. We do not attempt here to list the wealth of important problems in applied and experimental computational geometry now being addressed by the community as it responds to the application-driven need for practical geometric algorithms; we hope that an ongoing project to compile a more comprehensive list will address this omission. We encourage correspondence to correct, extend, and update a Web version of this list. 1. Can a minimum weight triangulation of a planar point set—one minimizing the total edge length— be found in polynomial time? This problem is one of the few from Garey and Johnson [GJ79] whose complexity status remains unknown. The best approximation algorithms achieve a (large) constant times the optimal length [LK96]; good heuristics are known [DMM95]. If Steiner points are allowed, again constant-factor approximations are known [Epp94, CL98], but it is open to decide even if a minimumweight Steiner triangulation exists (the minimum might be approached only in the limit). 2. What is the maximum number of combinatorial changes possible in a Euclidean Voronoi diagram of a set of n points each moving along a line at unit speed in two dimensions? The best lower bound known is quadratic, and the best upper bound is cubic [SA95, p. 297]. If the speeds are allowed to differ, the upper bound remains essentially cubic [AGMR98]. The general belief is that the complexity should be close to quadratic; Chew [Che97] showed this to be the case if the underlying metric is L1 (or L∞). 3. What is the combinatorial complexity of the Voronoi diagram of a set of lines (or line segments) in three dimensions? This problem is closely related to the previous problem, because points moving in the plane with constant velocity yield straight-line trajectories in space-time. Again, there is a gap between a lower bound of Ω(n) and an upper bound that is essentially cubic [Sha94] for the Euclidean case (and yet is quadratic for polyhedral metrics [BSTY98]). A recent advance shows that the “level sets” of the Voronoi diagram of lines, given by the union of a set of cylinders, indeed has near-quadratic complexity [AS00b]. 4. What is the complexity of the union of “fat” objects in R? The Minkowski sum of polyhedra of n vertices has complexity O(n) [AS99], as does the union of n congruent cubes [PSS01]. It is widely believed the same should hold true for fat objects, those with a bounded ratio of circumradius to inradius, as in does in R [ES00]. Dept. of Applied Mathematics and Statistics, University at Stony Brook, Stony Brook, New York 11794-3600, USA. jsbm@ ams.sunysb.edu. Supported by HRL Laboratories, NSF Grant CCR-9732221, NASA Ames Research Center, Northrop-Grumman Corporation, Sandia National Labs, and Sun Microsystems. Dept. of Computer Science, Smith College, Northampton, MA 01063, USA. [email protected]. Supported by NSF Grant CCR-9731804. 1 http://cs.smith.edu/~orourke/TOPP/.