A variational principle for circle packings. (Q1176645)

It was shown by \textit{P. Koebe} [Ber. Verh. Sächs. Akad. Wiss., Math.- Phys. Kl. 88, 141--164 (1936; Zbl 0017.21701; JFM 62.1217.04)] and rediscovered by \textit{E. M. Andreev} [Mat. Sb., Nov. Ser. 81 (123), 445--478 (1970; Zbl 0194.23202)]; ibid. 83 (125), 256--260 (1970; Zbl 0203.54904)] that any triangulated planar graph can be represented by nonoverlapping circular disks in the plane such that any two of them touch if and only if the corresponding vertices are adjacent. Furthermore, \textit{W. P. Thurston} [The Geometry and Topology of Three-Manifolds, Princeton Notes (1985), Chapter 13] showed that this representation is essentially unique. In the present paper (and the author's earlier paper in Forum Math. 1, No. 4, 395--402 (1989; Zbl 0685.52012)) the same problem, and some generalizations, are considered from an algorithmic point of view.