Constraint Satisfaction Programming for the No-three-in-line Problem

Using a constraint satisfaction approach, we exhibit configurations of $2n$ points on the $n\times n$ grid for all $n\le60$ with no three collinear. Consequently, the smallest $n$ for which it is unknown whether $D(n)=2n$ increases from $47$ to $61$.

💡 Research Summary

The paper revisits the classic “no‑three‑in‑line” problem, which asks for the largest possible number D(n) of lattice points that can be selected from an n × n grid without any three lying on a common straight line. The trivial upper bound is D(n) ≤ 2n, and probabilistic arguments suggest that for large n the maximum should be about (π/√3)·n ≈ 1.81 n. Prior computational work had established D(n)=2n for n ≤ 46 and for the isolated even values n = 48, 50, 52, but the status for larger n remained open.

The authors formulate the problem as a binary integer program. For each grid point (i,j) they introduce a 0/1 variable x_{ij} indicating occupancy. The core constraint is that for every affine line ℓ that contains at least three grid points, the sum of x_{ij} over ℓ∩G_n must be at most two. In addition, to achieve the bound 2n they enforce exactly two occupied points per row, i.e. Σ_j x_{ij}=2 for each i. This yields the feasibility model F_n.

Because the naïve model contains n² binary variables and Θ(n⁴) line constraints (asymptotically |L_n^{≥3}|≈5/12·π²·n⁴), the authors exploit the natural 90° rotational symmetry of many known solutions. They define the rotation ρ(i,j) = (j, n‑1‑i) and consider the cyclic group ⟨ρ⟩ (or ⟨ρ²⟩ for odd n). By restricting attention to a fundamental domain H_n (roughly one quarter of the grid) they introduce orbit variables y_{ij} for (i,j)∈H_n and link all rotated copies to the same y_{ij}. For even n every orbit has size four; for odd n diagonal points form size‑two orbits and the anti‑diagonal is forced empty. This symmetry reduction cuts the number of variables by a factor close to four and similarly reduces the number of line constraints after deduplication of identical incidence vectors.

The reduced model F_sym_n consists of binary variables y_{ij}, row‑sum constraints Σ_{(i,j)∈H_n} d_r(i,j)·y_{ij}=2 (ensuring two points per row), and line constraints Σ_{(i,j)∈H_n} c_ℓ(i,j)·y_{ij}≤2 for a representative set of lines L_{n,Γ}^{≥3}. The authors generate all line constraints, hash them to eliminate duplicates, and discard those that become tautological after symmetry reduction (e.g., lines intersecting the fixed anti‑diagonal in at most two free orbit points).

Implementation uses Google OR‑Tools’ CP‑SAT solver on a 192‑core AMD EPYC 9965 machine (384 hardware threads, 2.2 TB RAM). For each grid size n they launch M = 384 independent solver instances, each with a fresh random seed and a single search worker. The experiment stops as soon as any instance finds a feasible solution; the reported wall‑clock time is the minimum of the M runtimes. This “run‑until‑first‑success” strategy leverages the heavy‑tailed runtime distribution of CP‑SAT.

Empirical results show that without symmetry the solver reaches solutions only up to about n = 20 within an hour. With the symmetry‑reduced model, solutions up to n = 50 are found in roughly one hour, and n = 60 is solved in under 15 minutes on average (the fastest among the 384 parallel runs). Table 1 in the paper lists the representatives of the occupied orbits for each n from 47 to 60; applying the appropriate rotations reconstructs the full 2n‑point configurations. The authors also compare these findings with a probabilistic counting heuristic from Erdős–Turán type arguments, which predicts that the expected number of 2n‑point solutions drops below one around n ≈ 493. The fact that solutions exist for n = 60, and that incorporating symmetry into the heuristic makes the expected count drop below one already at n≈180, highlights the heuristic’s limitations.

Consequently, the authors prove that D(n)=2n for all 2 ≤ n ≤ 60, extending the known range and moving the smallest unresolved n from 47 to 61. The work demonstrates that careful symmetry exploitation combined with modern CP‑SAT technology can push exact combinatorial geometry results far beyond what was previously achievable. The paper concludes by suggesting future directions such as exploring larger n, testing other symmetry groups (e.g., 180° rotation), and benchmarking CP‑SAT against mixed‑integer programming or SAT‑based approaches for this and related lattice‑point problems.