[論文レビュー] Shadoks Approach to Parallel Reconfiguration of Triangulations

Guilherme D. da Fonseca, Fabien Feschet+1 more|arXiv (Cornell University)|Mar 22, 2026

Computational Geometry and Mesh Generation被引用数 0

ひとこと要約

The Shadoks team combines exact SAT-based methods with greedy heuristics and MaxSAT to solve the parallel reconfiguration of planar triangulations, winning CG:SHOP 2026 with provably optimal results on many instances.

ABSTRACT

We describe the methods used by Team Shadoks to win the CG:SHOP 2026 Challenge on parallel reconfiguration of planar triangulations. An instance is a collection of triangulations of a common point set. We must select a center triangulation and find short parallel-flip paths from each input triangulation to the center, minimizing the sum of path lengths. Our approach combines exact methods based on SAT with several greedy heuristics, and also makes use of SAT and MaxSAT for solution improvement. We present a SAT encoding for bounded-length paths and a global formulation for fixed path-length vectors. We discuss how these components interact in practice and summarize the performance of our solvers on the benchmark instances.

研究の動機と目的

• Motivate and address the CG:SHOP 2026 challenge on parallel reconfiguration of planar triangulations.
• Develop exact and heuristic techniques to minimize total path length across multiple input triangulations.
• Leverage SAT, MaxSAT, and greedy methods to compute optimal or near-optimal reconfigurations.
• Investigate the structure of solution spaces and the impact of conjectures on solvability.

提案手法

• Define unit and parallel flips and model reconfiguration as a path problem between triangulations.
• Formulate decision problems as SAT with edge and flip variables, enabling path existence checks.
• Introduce a path SAT formulation to decide if a length-l path exists between triangulations.
• Compute a lower bound via a cycle packing on a complete graph of triangulations, reducible to MaxSAT.
• Construct an exact solver that first computes a lower bound and then enumerates feasible length vectors using SAT, aided by backtracking.
• Explore the Happy Edges conjecture to improve pruning and solution quality, reducing search space.

Figure 2 : Illustration of a flip and the associated variable $f(u,v,u_{2},v_{2},i)$ .

実験結果

リサーチクエスチョン

• RQ1 Can a set of paths exist that reconfigure given input triangulations to a common center with minimal total length?
• RQ2 How can SAT/MaxSAT formulations be used to obtain exact solutions for parallel reconfiguration of triangulations?
• RQ3 What lower bounds can be derived for the objective, and how tight are they in practice?
• RQ4 Do properties like Happy Edges hold for parallel flips, and can they improve practical solvers?
• RQ5 What are the practical limits (instance size, number of triangulations) where the exact solver remains feasible?

主な発見

• The team won first place among 28 participating teams in CG:SHOP 2026.
• Their best solution achieved the best objective on 249 of 250 instances.
• They produced provably optimal solutions on 189 instances.
• The approach relies on CaDiCal (SAT) and EvalMaxSAT (MaxSAT) with additional heuristics and local search for improvement.
• The combination of exact SAT-based methods and greedy heuristics yielded strong performance across random, woc, and rirs instance classes.
• Experiments suggest the Happy Edges conjecture holds for parallel flips in their setting, enabling effective pruning.

Figure 3 : (a) A cycle packing. (b) Illustration of the proof. In this example, $d_{1,2}\leq r_{1}+r_{2}$ , $d_{2,1}\leq r_{2}+r_{1}$ , $d_{3,4}\leq r_{3}+r_{4}$ , $d_{4,5}\leq r_{4}+r_{5}$ , and $d_{5,3}\leq r_{5}+r_{3}$ by triangle inequality.