Approximate Shortest Paths in Polygons with Violations

Abstract

We study the problem of computing shortest k-violation path problem on polygons. Let P be a simple polygon in \(\mathbb {R}^2\) with n vertices and let s, t be a pair of points in P. Let int(P) represent the interior of P. In other words, \(int(P) = P \setminus \varDelta (P)\), where \(\varDelta (P)\) is the boundary of P. Let \(\tilde{P}=\mathbb {R}^2 \setminus int(P)\) represent the exterior of P. For an integer \(k \ge 0\), the problem of k-violation shortest path in P is the problem of computing the shortest path from s to t in P, such that at most k path segments are allowed to be in \(\tilde{P}\). The path segments are not allowed to bend in \(\tilde{P}\). For any k, we present a \((1+\epsilon )\) factor approximation algorithm for the problem, that runs in \(O(n^2 \sigma ^2 k\log n^2 \sigma ^2 + n^2 \sigma ^2 k)\) time. Here \(\sigma =O( \log _\delta (\frac{|L|}{r}))\) and \(\delta \), L, r are geometric parameters.