By Bloch S.J., et al. (eds.)
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Additional resources for Applications of algebraic K-theory to algebraic geometry and number theory, Part 1
We conclude that our computation times outperform previous approaches based on LP relaxations. With respect to using SDP, the computation times reported in  for the branch-and-bound algorithm based on SDP relaxations seem to be better while our method needs fewer subproblems. In general LP relaxations can be solved much faster than SDP relaxations. Consequently the method described in  needs to solve fewer relaxations. F. Anjos et al. Table 2 Root bounds, number of subproblems and CPU time to solve instances reported in  without target cut separation (N), with random (R) and with greedy (G) projections.
Furthermore, the Voronoi tiling obtained in this way is going to be a normal one because each Voronoi cell is contained in the closed ball of radius R concentric to the unit ball of the given Voronoi cell and therefore the diameter of each Voronoi cell is at most 2R. , it is at least 16:6508 : : : . After a sequence of partial results obtained in [3, 6], and  (proving the lower bounds 16:1433 : : : , 16:1445 : : : , and 16:1977 : : : ), just very recently, Hales  has announced a computer-assisted proof of the strong dodecahedral conjecture.
A. a. a. a. a. a. a. 543 632 713 observed that the bounds are very similar (cf. Sect. 3). Therefore we suspect that better computation times can be explained by stronger inequalities that were separated due to the different fractional points given by the interior-point method. In Table 3 we give the bounds at the root node, the number of subproblems and the CPU time needed by our branch-and-cut algorithm with and without switched clique separation for instances from , including some instances that were not reported in .