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Shadoks Approach to Parallel Reconfiguration of Triangulations

Guilherme D. da Fonseca, Fabien Feschet, Yan Gerard

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.

Shadoks Approach to Parallel Reconfiguration of Triangulations

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.
Paper Structure (24 sections, 6 theorems, 6 equations, 10 figures, 1 table)

This paper contains 24 sections, 6 theorems, 6 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Exact Algorithms
  3. Path SAT Formulation
  4. Start and target.
  5. Flips need edges.
  6. Flips keep edges.
  7. Flips flip edges.
  8. Edge changes require flips.
  9. Eliminating variables and clauses.
  10. Solution SAT Formulation
  11. Lower Bound
  12. The Exact Solver
  13. Heuristics
  14. Happy Edges Conjecture
  15. Crossing Lower Bound
  16. ...and 9 more sections

Key Result

Theorem 1

Consider two triangulations $T,T'$ of $S$ such that a parallel flip transforms $T$ into $T'$ and a segment $s$ with endpoints in $S$. Let $\chi,\chi'$ respectively denote the number of edges of $T,T'$ crossed by $s$. We then have $\chi' \geq \lfloor \chi / 2 \rfloor$.

Figures (10)

  • Figure 1: Our best centers to instances random_78_40_10, woc-70-random-9a7d18d3, woc-90-tsplib, and rirs-500-50-23d00ec5, respectively.
  • Figure 2: Illustration of a flip and the associated variable $f(u,v,u_2,v_2,i)$.
  • 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.
  • Figure 4: A path of length $3$ to insert an edge that had $6$ crossings.
  • Figure 5: A path of length $4$ to insert an edge that had $6$ crossings. Each triangle is labeled and colored as containing an upper or a lower edge.
  • ...and 5 more figures

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • Conjecture 3
  • Theorem 4
  • Corollary 5
  • Corollary 6
  • Theorem 7