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Generalized Hi-Q is NP-complete

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Abstract

This paper deals with a popular puzzle known as Hi-Q. The puzzle is generalized: the board is extended to the size n × n, an initial position of the puzzle is given, and a place is given on which only one token is finally placed. The complexity of the generalized Hi-Q is proved NP-complete.
Japan Advanced Institute of Science and Technology
JAIST Repository
https://dspace.jaist.ac.jp/
Title Generalized Hi-Q is NP-Complete
Author(s) UEHARA, Ryuhei; IWATA, Shigeki
Citation The TRANSACTIONS of the IEICE, E73-E(2): 270-273
Issue Date 1990-02-20
Type Journal Article
Text version publisher
URL http://hdl.handle.net/10119/4709
Rights
Copyright (C)1990 IEICE. Ryuhei Uehara and
Shigeki Iwata, The TRANSACTIONS of the IEICE,
E73-E(2), 1990, 270-273.
http://www.ieice.org/jpn/trans_online/
Description
... This property can be found in many natural puzzles and hence this fact has led us to many results of NPcompleteness of traditional, classic, and familiar puzzles. For example, the following puzzles and card games are all NP-complete problems, and most results had obtained in the early stage in this context: Clicomania (also known as (a.k.a.) Same Game) [9], peg solitaire [101] (see Fig. 3), cryptarithm [28] (see Fig. 4), Klondike [78], FreeCell [46], jigsaw puzzle [19,39], and packing puzzles [8,19,53,83,91]. ...
Article
Since the 1930s, mathematicians and computer scientists have been interested in computation. While mathematicians investigate recursion theory, computer scientists investigate computational complexity based on Turing machine model to understand what a computation is. Beside them, there is another approach of research on computation, which is the investigation of puzzles and games. Once we regard the rules used in puzzles and games as the set of basic operations of computation, we can perform some computation by solving puzzles and playing games. In fact, research on puzzles and games from the viewpoint of theoretical computer science has continued without any break in the history of theoretical computer science. Sometimes the research on computational complexity classes has proceeded by understanding the tons of puzzles. The wide collection of complete problems for a specific computational complexity class shares a common property, which gives us a deep understanding of the class. In this survey paper, we give a brief history of research on computational complexities of puzzles and games with related results and trends in theoretical computer science.
... https://www.thinkfun.com/products/rush-hour/ taire) [10]. In these games, solvers must manipulate physical pieces to solve a challenge. ...
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Challenges for physical solitaire puzzle games are typically designed in advance by humans and limited in number. Alternately, some games incorporate stochastic setup rules, where the human solver randomly sets up the game board before solving the challenge, which can greatly increase the number of possible challenges. However, these setup rules can often generate unsolvable or uninteresting challenges. To better understand these setup processes, we apply a taxonomy for procedural content generation algorithms to solitaire puzzle games. In particular, for the game Fujisan, we examine how different stochastic challenge generation algorithms attempt to minimize undesirable challenges, and we report their affect on ease of physical setup, challenge solvability, and challenge difficulty. We find that algorithms can be simple for the solver yet generate solvable and difficult challenges, by constraining randomness through embedding sub-elements of the puzzle mechanics into the physical pieces of the game.
... They are actually still a good testbed for artificial intelligence techniques [22,21]. Indeed, the first computational hardness result for this game was proved in 1990 in [25], but was limited to those peg solitaire puzzles in which the final configuration is required to have only one peg (and hence the goal was that of cleaning the entire board). The Solitaire-Army problem has received a lot of attention from the mathematical and game-connoisseur community [8,1,9]. ...
Article
Full-text available
Despite its long history, the classical game of peg solitaire continues to attract the attention of the scientific community. In this paper, we consider two problems with an algorithmic flavour which are related with this game, namely Solitaire-Reachability and Solitaire-Army. In the first one, we show that deciding whether there is a sequence of jumps which allows a given initial configuration of pegs to reach a target position is NP-complete. Regarding Solitaire-Army, the aim is to successfully deploy an army of pegs in a given region of the board in order to reach a target position. By solving an auxiliary problem with relaxed constraints, we are able to answer some open questions raised by Cs\'ak\'any and Juh\'asz (Mathematics Magazine, 2000).
Article
Peg solitaire is a single-player board game. The goal of the game is to remove all but one peg from the game board. Peg solitaire on graphs is a peg solitaire played on arbitrary graphs. A graph is called solvable if there exists some vertex s such that it is possible to remove all but one peg starting with s as the initial hole. In this paper, we prove that it is NP-complete to decide if a graph is solvable or not.
Chapter
In combinatorial reconfiguration, the reconfiguration problems on a vertex subset (e.g., an independent set) are well investigated. In these problems, some tokens are placed on a subset of vertices of the graph, and there are three natural reconfiguration rules called “token sliding,” “token jumping,” and “token addition and removal”. In the context of computational complexity of puzzles, the sliding block puzzles play an important role. Depending on the rules and set of pieces, the sliding block puzzles characterize the computational complexity classes including P, NP, and PSPACE. The sliding block puzzles correspond to the token sliding model in the context of combinatorial reconfiguration. On the other hand, a relatively new notion of jumping block puzzles is proposed in puzzle society. This is the counterpart to the token jumping model of the combinatorial reconfiguration problems in the context of block puzzles. We investigate several variants of jumping block puzzles and determine their computational complexities.
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We consider the popular smartphone game Trainyard: a puzzle game that requires the player to lay down tracks in order to route colored trains from departure stations to suitable arrival stations. While it is already known [Almanza et al., FUN 2016] that the problem of finding a solution to a given Trainyard instance (i.e., game level) is NP-hard, determining the computational complexity of checking whether a candidate solution (i.e., a track layout) solves the level was left as an open problem. In this paper we prove that this verification problem is PSPACE-complete, thus implying that Trainyard players might not only have a hard time finding solutions to a given level, but they might even be unable to efficiently recognize them.
Chapter
We discuss some results on the complexity of games and puzzles. In particular, we focus on the relationship between bounded treewidth and the (in-)tractability of games and puzzles in which graphs play a role. We discuss some general methods which are good starting points for finding complexity proofs for games and puzzles.
Article
Challenges for physical solitaire puzzle games are typically designed in advance by humans and limited in number. Alternatively, some games incorporate rules for stochastic setup, where the human solver randomly sets up the game board before solving the challenge. These setup rules greatly increase the number of possible challenges, but can often generate unsolvable or uninteresting challenges. To better understand the compromises involved in minimizing undesirable challenges, we examine three games where component design choices can influence the stochastic nature of the resulting challenge generation algorithms. We evaluate the effect of these components and algorithms on challenge solvability and challenge engagement. We find that algorithms which control randomness through entangling components based on sub-elements of the puzzle mechanics can generate interesting challenges with a high probability of being solvable.
Article
Sokoban is one of the most studied combinatorial puzzle game in the literature. Its computational complexity was first shown to be PSPACE-complete in 1997. In 2005, a new proof of this result was obtained by Hearn and Demaine, by introducing the Nondeterministic Constraint Logic (Ncl) problem [R. A. Hearn and E. D. Demaine, PSPACE-completeness of sliding-block puzzles and other problems through the nondeterministic constraint logic model of computation, Theoretical Computer Science 343 (2005) 72-96]. Since then, Ncl has been used to prove the PSPACE-completeness of several other puzzles including a few Sokoban variants, by many authors. In this paper, we show that Snowman, a new Sokoban-like puzzle game released in 2015, is PSPACE-complete by reduction from Ncl.
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