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**1 - 2**of**2**### Cluster Connecting Problem inside a Polygon

"... The cluster connecting problem inside a simple polygon is introduced in this paper. The problem is shown to be NP-complete. A log n-factor approximation algorithm is proposed. 1 ..."

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The cluster connecting problem inside a simple polygon is introduced in this paper. The problem is shown to be NP-complete. A log n-factor approximation algorithm is proposed. 1

### A Heuristic Homotopic Path Simplification Algorithm

"... Abstract. We study the well-known problem of approximating a polygonal path P by a coarse one, whose vertices are a subset of the vertices of P. In this problem, for a given error, the goal is to find a path with the minimum number of vertices while preserving the homotopy in presence of a given set ..."

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Abstract. We study the well-known problem of approximating a polygonal path P by a coarse one, whose vertices are a subset of the vertices of P. In this problem, for a given error, the goal is to find a path with the minimum number of vertices while preserving the homotopy in presence of a given set of extra points in the plane. We present a heuristic method for homotopy-preserving simplification under any desired measure for general paths. Our algorithm for finding homotopic shortcuts runs in O(m log(n + m) +n log n log(nm)+k) time, where k is the number of homotopic shortcuts. Using this method, we obtain an O(n 2 + m log(n + m)+n log n log(nm)) time algorithm for simplification under the Hausdorff measure.