ETH Flippers Approach to Parallel Reconfiguration of Triangulations: SAT formulation and Heuristics

We describe the algorithms used by the ETH Flippers team in the CG:SHOP 2026 Challenge. Each instance consists of a set of triangulations on a common point set, and the objective is to find a central triangulation that minimizes the total parallel flip distance to the input set. Our strategy combines an exact solver for small and medium-sized instances with a suite of heuristics for larger instances. For the exact approach, we formulate the problem as a SAT instance with XOR clauses to model edge transitions across multiple rounds, further optimized by lower bounds derived from exact pairwise distances. For larger instances, we use a greedy local search and edge-coloring techniques to identify maximal sets of independent flips. Our approach ranked second overall and first in the junior category, computing provably optimal solutions for 186 out of 250 instances.

💡 Research Summary

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The paper presents the solution developed by the ETH Flippers team for the CG:SHOP 2026 Challenge, which asks participants to find a central triangulation C for a given set of triangulations {T₁,…,Tₘ} on a common point set P such that the sum of parallel flip distances from each Tᵢ to C is minimized. The authors combine two complementary strategies: an exact SAT‑based solver for small and medium instances, and a suite of heuristics for large instances.

Exact SAT formulation
For instances with up to about 320 points, the problem is encoded as a SAT instance enriched with XOR clauses. The key idea is to model the state of each edge across a sequence of parallel flip rounds. For an edge e, its initial presence e₀ and final presence eₖ are linked by an XOR of all flip variables that involve e as a diagonal. This captures the fact that an edge toggles exactly when a flip on a quadrilateral containing e is performed. Global variables c_uv represent the edges of the central triangulation; a final‑step equivalence forces eₖ ⇔ c_uv, thereby ensuring that all input triangulations converge to the same C.

To prevent illegal simultaneous flips, the authors add at‑most‑one (AMO) constraints for every set of flips that share a vertex quadruple. They implement AMO using ladder encoding (2r − 1 auxiliary variables and 3r − 2 clauses for a set of size r), which is more compact than the naïve pairwise encoding.

A crucial preprocessing step computes lower bounds on the earliest round at which each edge can appear in a triangulation, using a breadth‑first search on the flip graph. Variables and clauses that would involve an edge before its theoretical minimum round are omitted, dramatically shrinking the search space. Convex‑hull edges are fixed throughout all rounds, further pruning the model.

The SAT model is fed to CryptoMiniSat 5, a solver that natively handles XOR clauses, allowing the exact solver to handle instances with up to 320 points and total parallel flip distance 92. For 186 out of 250 benchmark instances the solver proved optimality; three additional instances were solved but a post‑processing bug prevented the optimal value from being recorded.

Finding empty convex quadrilaterals
The SAT encoding requires the set Q of all empty convex quadrilaterals of P, which can be Θ(n⁴) in the worst case. The authors evaluate two spatial indexing techniques. Although k‑d trees offer O(√N) worst‑case range‑query time, they incur pointer‑chasing overhead and are unnecessary for the quasi‑uniform point distributions of the benchmark. Instead, a uniform grid (spatial hashing) is built: the bounding box of P is divided into N cells, each cell expected to contain O(1) points. For each candidate quadrilateral, its axis‑aligned bounding box is computed, and only the intersecting grid cells are inspected for interior points. This yields expected O(1) query time and excellent cache locality, making the preprocessing step fast enough for the competition.

Heuristic approach for large instances
When the SAT model becomes too large (instances with many points or high flip distances), the team switches to a heuristic pipeline. The pipeline consists of:

1. Greedy local search – Starting from an initial central triangulation (often one of the input triangulations), the algorithm repeatedly selects edges whose flip would most reduce the total distance, flips them, and updates the distance estimates. The search is limited to a small number of rounds (typically 10–20) to keep runtime low.

2. Edge‑coloring for independent flips – In each round the algorithm builds a conflict graph where vertices are potential flips and edges indicate that the two flips share a triangle. A graph‑coloring (or maximal independent set) algorithm is applied so that all flips of the same color can be performed simultaneously. This maximizes the number of parallel flips per round, mimicking the optimal parallel schedule.

3. Combination of the two – The heuristic first applies large independent flip sets to quickly reduce the bulk of the distance, then refines the solution with the greedy local search to handle remaining mismatches.

The heuristic consistently produces solutions within a few percent of the optimal values on the large benchmark instances, and it was responsible for the team’s overall second place finish.

Experimental results

• Benchmark: 250 instances, point counts 100–320, triangulation counts 160–500.
• Exact solver: proved optimality for 186 instances, unique optimal solution for the hardest instance (random_instance_440_160_20) with total distance 92.
• Heuristic: solved the remaining instances with sub‑optimal but competitive values.
• Overall ranking: 2nd place overall, 1st place in the junior category.

Conclusions and future work
The paper demonstrates that a hybrid approach—exact SAT encoding with XOR clauses for small instances and a well‑designed heuristic for large instances—can effectively tackle a challenging geometric optimization problem. The XOR‑based round‑by‑round modeling dramatically reduces the number of SAT variables compared with a pure CNF formulation, while the spatial‑hashing preprocessing makes the enumeration of empty quadrilaterals practical. Future directions include extending the SAT model to multi‑objective settings (e.g., minimizing a weighted sum of total flips and number of rounds), integrating more sophisticated meta‑heuristics (genetic algorithms, simulated annealing) into the local search, and benchmarking against newer SAT solvers that support native cardinality and pseudo‑boolean constraints.