Geometric Bicriteria Optimal Path Problems

A bicriteria optimal path simultaneously satisfies two bounds on two measures of path quality. The complexity of finding such a path depends on the particular choices of path quality. This thesis studies bicriteria path problems in a geometric setting using several pairs of path quality, including: path length measured according to different norms $(L\sb{p}$ and $L\sb{q});$ Euclidean length within two or more classes of regions; total turn and Euclidean length; total turn and number of links; and Euclidean length and number of links. For several cases, finding the bicriteria optimal path is shown to be NP-hard. These NP-hard cases include minimizing path length in two different norms, minimizing travel through two regions, and minimizing length and total turn. In the last case, an $O(En\sp2N\sp2$) pseudo-polynomial time algorithm to find an approximate answer is presented. In contrast, when the two measures of path quality are total turn and number of links, an $O(E\sp3n$log$\sp2n)$ exact algorithm is given. A main result of this thesis examines minimizing the Euclidean length and number of links of a path. When the geometric setting of this problem is a polygon without holes, this thesis presents an $O(n\sp3k\sp3$log$(Nk/\epsilon\sp{1/k}))$ algorithm to find a k-link path with Euclidean length at most 1 + $\epsilon$ times the length of the shortest k-link path. A faster algorithm for a relaxed case, when the output path is allowed to have 2k links, is presented for a polygon with or without holes. Finally, some approximation algorithms are outlined for finding a minimum link path among polyhedral obstacles.