Optimal grid-size ratio for adjacency-constrained world map embeddings

Determine whether there exist world map constructions on a K×K grid with ratio R = K/N strictly below 1.5 for the adjacency-constrained labeling problem (given N countries and edge set E of required adjacencies), and ascertain the optimal ratio R as a function of the adjacency graph structure (i.e., for different patterns of adjacency constraints).

Background

The "World Map" task asks for a K×K grid whose cell labels from {1,…,N} satisfy adjacency constraints defined by an undirected edge set E: every required pair must appear in at least one pair of orthogonally adjacent cells, and any orthogonally adjacent cells must correspond to an edge in E. The grading focuses on compactness by minimizing K and measuring R = K/N. A prior IOI submission achieves R' = 1.5 across inputs, establishing a strong baseline.

The paper explicitly states uncertainty about the existence of better-than-1.5 constructions and about the optimal achievable ratio across different adjacency patterns, framing a concrete open question about grid compactness under these adjacency constraints.