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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 $23$.

To study the fixed parameter undecidability of tiling problem for a set of Wang tiles, Jeandel and Rolin show that the tiling problem for a set of 44 Wang bars is undecidable. In this paper, we improve their result by proving that whether a set of 29 Wang bars can tile the plane is undecidable. As a consequence, the tiling problem for a set of Wang tiles with color deficiency of 25 is also undecidable.

We give a proof of Ollinger's conjecture that the problem of tiling the plane with translated copies of a set of $8$ polyominoes is undecidable. The techniques employed in our proof include a different orientation for simulating the Wang tiles in polyomino and a new method for encoding the colors of Wang tiles.

Burke and Teng introduced a two-player combinatorial game Atropos based on Sperner's lemma, and showed that deciding whether one has a winning strategy for Atropos is PSPACE-complete. In the original Atropos game, the players must color a node adjacent to the last colored node. Burke and Teng also mentioned a variant Atropos-k in which each move is at most of distance k of the previous move, and asked a question on determining the computational complexity of this variant. In this paper, we answer this question by showing that for any fixed integer k (k>=2), Atropos-k is PSPACE-complete by reduction from True Quantified Boolean Formula (TQBF).

In this paper, we show that the friends-and-strangers problem is PSPACE-complete by reduction from the Ncl (non-deterministic constraint logic) problem.

We consider the computational complexity of Hearthstone which is a popular online CCG (collectible card game). We reduce a PSPACE-complete problem, the partition game, to perfect information Hearthstone in which there is no hidden information or random elements. In the reduction, each turn in Hearthstone is used to simulate one choice in the partition game. It is proved that determining whether the player has a forced win in perfect information Hearthstone is PSPACE-hard., Comment: 18 pages, 6 tables

A selfmate is a Chess problem in which White, moving first, needs to force Black to checkmate within a specified number of moves. The reflexmate is a derivative of the selfmate in which White compels Black to checkmate with the added condition that if either player can checkmate, they must do that (when this condition applies only to Black, it is a semi-reflexmate). We slightly modify the reduction of EXPTIME-hardness of Chess and apply the reduction to these Chess problems. It is proved that selfmate, reflexmate, and semi-reflexmate are all EXPTIME-complete., Comment: 12 pages, 7 figures

Kill-all Go is a variant of Go in which Black tries to capture all white stones, while White tries to survive. We consider computational complexity of Kill-all Go with two rulesets, Chinese rules and Japanese rules. We prove that: (i) Kill-all Go with Chinese rules is PSPACE-hard, and (ii) Kill-all Go with Japanese rules is EXPTIME-complete., Comment: 15 pages, 11 figures

Dou Shou Qi is a Chinese strategy board game for two players. We use a EXPTIME-hardness framework to analyse computational complexity of the game. We construct all gadgets of the hardness framework. In conclusion, we prove that Dou Shou Qi is EXPTIME-complete., Comment: 11 pages, 11 figures. arXiv admin note: text overlap with arXiv:1904.00200

We review NP-hardness framework and PSPACE-hardness framework for a type of 2D platform games. We introduce a EXPTIME-hardness framework by defining some new gadgets. We use these hardness frameworks to analyse computational complexity of Xiangqi (Chinese Chess) and Janggi (Korean Chess). We construct all gadgets of the hardness frameworks in Xiangqi and Janggi. In conclusion, we prove that Xiangqi and Janggi are both EXPTIME-complete., Comment: 41 pages, 34 figures

We apply symbolic method to deduce functional equation which generating function of counting sequence of dependency trees must satisfy. Then we use Lagrange inversion theorem to obtain concrete expression of the counting sequence. We apply the famous Stirling's approximation to get approximation of the counting sequence. At last, we discuss the additive parameters of dependency trees., Comment: 4 pages

For complexity of the heterogeneous minimum spanning forest problem has not been determined, we reduce 3-SAT which is NP-complete to 2-heterogeneous minimum spanning forest problem to prove this problem is NP-hard and spread result to general problem, which determines complexity of this problem. It provides a theoretical basis for the future designing of approximation algorithms for the problem., Comment: 3 pages, 1 figure