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266 Yinfeng Xu and Wenqiang DaiThe fact that E 4 (S) is the subgraph of MW T (S), but no algorithm is found to computeE 4 (S) in polynomial time, leads to the following problem.Old Problem 5: Is it possible to find the intersection of all local optimal triangulations in a polynomialtime?For a given edge e, we may ask a question as whether there is a local optimal triangulationT L (S) such that e is an edge of T L (S). To observe an edge e in E 4 (S), we may know that e isin any local optimal triangulation of S and for any edge e ′ with endpoints in S interests e, e ′can not be in any local optimal triangulation. Let E ′ 4 (S) denote the set of edges do not in any4-optimal triangulation for point set S, and E4 ∗ (S) = {e | there is a local optimal triangulationT (S) such that e ∈ T (S)}. From the definition of E 4 (S), E ′ 4 (S) and E∗ 4 (S), we haveE(S) = E ′ 4(S) ∪ E ∗ 4(S), E 4 (S) ⊂ E ′ 4(S), E 4 (S) ∩ E ′ 4(S) = φ, E ′ 4(S) ∩ E ∗ 4(S) = φIf any one of E 4 (S), E ′ 4 (S), E∗ 4 (S) can be computed in polynomial time then we may computethe another two in polynomial time easily. So the following problem is interesting.New Problem 3: How to test whether there is a local optimal triangulation contains a given edge e?2.4 Pseudo-triangulationA pseudo-triangle is a simple polygon with exactly three vertices where the inner angel is lessthan π. A pseudo-triangulation of a point set S is a partition of the interior of the convex hull ofS into a set of pseudo-triangles.In many ways, pseudo-triangulations have nicer properties than classical triangulations ofa point set [2,19]. Among them, a minimum pseudo-triangulation of a points set is one with thesmallest possible number of edges and a minimum weight pseudo-triangulation (MW P T ) is apseudo-triangulation which minimizes the sum of the edge lengths. No result is known aboutthe complexity of computing a MW P T of a given point set S, and a similar problem arisesfor finding a pseudo-triangulation within a given triangulation.New Problem 4: Whether to find a MW P T is NPC?New Problem 5: How to find a MW P T as a subgraph of a given triangulation?2.5 Mesh GenerationGenerating triangular meshes is one of the fundamental problems in computational geometry,and has been extensively studied; see e.g. the survey article by Bern Eppstein [5]. In view ofthe field of application, it is quite natural to consider mesh generation problem under someoptimal criteria. Recently, we consider the problem of a mesh generation with some edgelength constraints. Among the most fascinating and challenging, we mention the following.New Problem 6: For given real numbers α ≤ β ≤ γ, and a convex polygon P , how to find a triangulation,T (P ), of P such that the inner edge length in T (P ) is in the interval [α, γ] and the number ofedges with edge length different from β is minimum?In [23], we have presented a heuristics to generate a triangular mesh for a special case ofthe above problem, but for a general case, this problem is still open.