Flip Distance Between Triangulations of a Simple Polygon is NP-Complete

Let T be a triangulation of a simple polygon. A flip in T is the operation of replacing one diagonal of T by a different one such that the resulting graph is again a triangulation. The flip distance between two triangulations is the smallest number of flips required to transform one triangulation into the other. For the special case of convex polygons, the problem of determining the shortest flip distance between two triangulations is equivalent to determining the rotation distance between two binary trees, a central problem which is still open after over 25 years of intensive study. We show that computing the flip distance between two triangulations of a simple polygon is NP-hard. This complements a recent result that shows APX-hardness of determining the flip distance between two triangulations of a planar point set.

[1]  Charles L. Lawson,et al.  Transforming triangulations , 1972, Discret. Math..

[2]  Robert E. Tarjan,et al.  Rotation distance, triangulations, and hyperbolic geometry , 1986, STOC '86.

[3]  K. N. Dollman,et al.  - 1 , 1743 .

[4]  J. Cremona,et al.  ELLIPTIC CURVES: (Graduate Texts in Mathematics 111) , 1990 .

[5]  Alexander Pilz.  Flip Distance Between Triangulations of a Planar Point Set is NP-Complete , 2012, ArXiv.

[6]  Thomas Ottmann,et al.  The Edge-flipping Distance of Triangulations , 1996, J. Univers. Comput. Sci..

[7]  J. Milne.  Elliptic Curves , 2020 .

[8]  Alexander Pilz,et al.  Flip Distance between Triangulations of a Simple Polygon is NP-Complete , 2013, ESA.

[9]  Raimund Seidel,et al.  Constructing arrangements of lines and hyperplanes with applications , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[10]  Prosenjit Bose,et al.  Every Large Point Set contains Many Collinear Points or an Empty Pentagon , 2009, CCCG.

[11]  Ge Xia,et al.  Flip Distance is in FPT time $O(n+ k \cdot c^k)$ , 2014, ArXiv.

[12]  Leonidas J. Guibas,et al.  The power of geometric duality , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[13]  Prosenjit Bose,et al.  Flips in planar graphs , 2009, Comput. Geom..

[14]  Marc Noy,et al.  Flipping Edges in Triangulations , 1996, SCG '96.

[15]  Derick Wood,et al.  A Note on Some Tree Similarity Measures , 1982, Inf. Process. Lett..

[16]  Vinayak Pathak,et al.  Flip Distance Between Two Triangulations of a Point Set is NP-complete , 2012, CCCG.

[17]  Hans Jürgen Prömel,et al.  The Steiner Tree Problem , 2002 .

[18]  Bruce Randall Donald,et al.  A rational rotation method for robust geometric algorithms , 1991, SCG '92.

[19]  Bing Lu,et al.  Polynomial Time Approximation Scheme for the Rectilinear Steiner Arborescence Problem , 2000, J. Comb. Optim..

[20]  Weiping Shi,et al.  The rectilinear Steiner arborescence problem is NP-complete , 2000, SODA '00.

[21]  C. Lawson.  Software for C1 Surface Interpolation , 1977 .

[22]  David Eppstein,et al.  Happy endings for flip graphs , 2006, SCG '07.

[23]  Alexander Pilz.  Flip distance between triangulations of a planar point set is APX-hard , 2014, Comput. Geom..

[24]  Frank K. Hwang,et al.  The rectilinear steiner arborescence problem , 2005, Algorithmica.

[25]  V. A. Trubin.  Subclass of the Steiner problems on a plane with rectilinear metric , 1985 .

[26]  Ge Xia,et al.  Flip Distance Is in FPT Time O(n+ k * c^k) , 2015, STACS.