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 $23$.

翻译：本文研究使用固定瓷砖集合铺砌有限单连通区域的计算复杂度。我们证明，使用固定23种王瓷砖铺砌单连通区域的问题是NP完全的。由此推论，使用固定111种矩形瓷砖铺砌单连通区域的问题也是NP完全的。我们的结果通过使用更少的瓷砖数量改进了Igor Pak与Jed Yang的研究。值得注意的是，在王瓷砖情况下，瓷砖数量从35种减少到23种，降幅超过三分之一。