NP-completeness of Tiling Finite Simply Connected Regions with a Fixed Set of Wang Tiles

TL;DR

It is shown that the problem of tiling simply connected regions with a fixed set of Wang tiles is NP-complete, and the results improve that of Igor Pak and Jed Yang by using fewer numbers of tiles.

Abstract

The computational complexity of tiling finite simply connected regions with a fixed set of tiles is studied in this paper. We show that the problem of tiling simply connected regions with a fixed set of $23$ Wang tiles is NP-complete. As a consequence, the problem of tiling simply connected regions with a fixed set of $111$ rectangles is NP-complete. Our results improve that of Igor Pak and Jed Yang by using fewer numbers of tiles. Notably in the case of Wang tiles, the number has decreased by more than one third from $35$ to $23$.

Figures (11)

• Figure 1: The variables ($V_0, V_1$), clauses ($C_0, C_1$), forwarders ($F_i, i=0,1$), left anchors ($L_i, i=0,1$) and right anchors ($R_i, i=0,1$), and crossovers ($X_{ij}, i,j=0,1$).
• Figure 2: The simply connected region corresponding to $\varphi$.
• Figure 3: The clause sub-regions.
• Figure 4: The crossover sub-region.
• Figure 5: The variables ($V_0, V_{1x}, V_{1y}$), clauses ($C_0, C_1$), forwarders ($F_i, i=0,1$), left anchors ($L_i, i=0,1$), right anchors ($R_i, i=0,1$), crossovers ($X_{ii}$ and $X_{i,1-i}, i=0,1$) (modified).

Theorems & Definitions (15)

• Theorem 1: Tiling with two bars bnrr95
• Theorem 2: y_thesis
• Theorem 3
• Definition 1: Tiling finite simply connected regions with a fixed set of tiles
• Definition 2: Cubic Monotone 1-in-3 SAT
• proof : Proof of Theorem \ref{['thm_main']}
• Lemma 1
• proof
• Lemma 2: y_thesis
• Theorem 4