A theoretical preprint reports exact algorithms for two planar geometry problems whose stated worst-case runtimes fall below the 2^n enumeration barrier. The optimal-discretization procedure is given a running time of O(1.9602^n). For point separation, treated as a single-set, axis-parallel variant, the bounds are O(1.8899^n) in general position and O(1.8906^n) when shared coordinates are allowed. The authors describe these as the first runtimes below 2^n for both problems.
The problem behind the numbers
The first problem takes two planar point sets, R and B, with n = |R| + |B| total points. The second studies point separation as a single-set, axis-parallel variant of the discretization problem. The running-time bounds are expressed in terms of the number of points in the input.
Where the speedup comes from
The algorithms first use structural balance observations to reduce the enumeration search space. Candidate solutions are then completed by treating the complementary separator problem as interval hitting, meaning a choice that intersects every interval in a collection. A greedy procedure solves that completion step optimally in polynomial time. The gain comes from shrinking the exponential part of the search before applying this exact final step.
Balance matters
The structural bounds are clearest in general position, the paper's case where shared-coordinate degeneracy is not present. For optimal discretization, an optimal solution exists in which the smaller of the horizontal and vertical line counts is no more than 0.4n. The point-separation bound is stronger: in general position, the smaller orientation has at most (n−1)/3 lines. These are existence guarantees, so they identify a solution with that balance rather than requiring every optimal solution to have the same shape.
Allowing shared coordinates changes the formulas. For optimal discretization, the paper gives an upper bound of 0.4n + 0.6d for the smaller orientation, where d = min{dx, dy} is its degeneracy parameter. For point separation, the corresponding bound is n/3 + 2d/3. The extra terms make explicit that shared-coordinate degeneracy affects the structural guarantee.
A sharper limit in one case
The optimal-discretization balance bound is not presented as the final word. The authors construct an infinite family of general-position instances in which every optimal solution must use at least 0.3n lines in its smaller orientation. Taken together with the 0.4n upper bound, the construction leaves a gap between what the analysis guarantees and what the examples force.
For point separation, the construction reaches the general-position threshold exactly. For every constructed instance with n = 3k + 1 points, every optimal solution has at least (n−1)/3 lines in its smaller orientation, matching the stated upper bound. The balance limit is therefore tight for that constructed family.
What the bounds mean
The runtime figures are theoretical guarantees for exact algorithms on abstract point-set inputs, not reported wall-clock measurements for a particular implementation. The work stays within axis-parallel geometry, so its statements concern that setting and do not by themselves extend to arbitrary slopes or rectangulations.
The manuscript also includes a disclosure about how it was prepared. The authors say they used ChatGPT 5.5 and Claude Opus 4.7 to explore the problem structure and verify arguments, while describing the writeup as their own and accepting full responsibility. The supplied metadata identifies the work as arXiv:2608.25197v1, a preprint dated 25 August 2026. No funding statement is reported in the supplied document or metadata.
Paper data and sources
Original title: Exact algorithms for optimal discretization
Authors: László Kozma, Junqi Tan
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
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