Tetris is Hard, Even to Approximate

Demaine, Erik D.; Hohenberger, Susan; Liben-Nowell, David

doi:10.1007/3-540-45071-8_36

Cited by 76 publications

(68 citation statements)

References 7 publications

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“…Finding the optimal placement of a given series of falling blocks is an NP-complete problem (Demaine et al, 2002). Although the optimal placement is not required to play a good game of Tetris, the added difficulty of dealing with an unknown series of blocks makes it quite challenging for reinforcement learning, and Q-learning in particular.…”

Section: Comparison With Previous Rrl-implementationsmentioning

confidence: 99%

Graph kernels and Gaussian processes for relational reinforcement learning

2006

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RRL is a relational reinforcement learning system based on Q-learning in relational state-action spaces. It aims to enable agents to learn how to act in an environment that has no natural representation as a tuple of constants. For relational reinforcement learning, the learning algorithm used to approximate the mapping between state-action pairs and their so called Q(uality)-value has to be very reliable, and it has to be able to handle the relational representation of state-action pairs. In this paper we investigate the use of Gaussian processes to approximate the Q-values of state-action pairs. In order to employ Gaussian processes in a relational setting we propose graph kernels as a covariance function between state-action pairs. The standard prediction mechanism for Gaussian processes requires a matrix inversion which can become unstable when the kernel matrix has low rank. These instabilities can be avoided by employing QR-factorization. This leads to better and more stable performance of the algorithm and a more efficient incremental update mechanism. Experiments conducted in the blocks world and with the Tetris game show that Gaussian processes with graph kernels can compete with, and often improve on, regression trees and instance based regression as a generalization algorithm for RRL.

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Section: Comparison With Previous Rrl-implementationsmentioning

confidence: 99%

Graph kernels and Gaussian processes for relational reinforcement learning

2006

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“…For example, Tetris [4], Solitaire [12], Minesweeper [11], (n × n)-extension of the 15-puzzle [16], the Slither Link Puzzle [19], and Sokoban (a transport puzzle in a maze) [5], Puyo puyo (also known as Puyo pop) [14], and Hashiwokakero [3] are known to be NP-hard. Uehara proved that deciding whether a given pop-up book can be opened (or closed) is NPhard [18].…”

Section: Introductionmentioning

confidence: 99%

NP-Hard and k-EXPSPACE-Hard Cast Puzzles

Iwamoto

Sasaki

Nishio

et al. 2010

IEICE Trans. Inf. & Syst.

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SUMMARYA disentanglement puzzle consists of mechanically interlinked pieces, and the puzzle is solved by disentangling one piece from another set of pieces. A cast puzzle is a type of disentanglement puzzle, where each piece is a zinc die-casting alloy. In this paper, we consider the generalized cast puzzle problem whose input is the layout of a finite number of pieces (polyhedrons) in the 3-dimensional Euclidean space. For every integer k ≥ 0, we present a polynomial-time transformation from an arbitrary k-exponential-space Turing machine M and its input x to a cast puzzle c 1 of size k-exponential in |x| such that M accepts x if and only if c 1 is solvable. Here, the layout of c 1 is encoded as a string of length polynomial (even if c 1 has size k-exponential). Therefore, the cast puzzle problem of size k-exponential is k-EXPSPACE-hard for every integer k ≥ 0. We also present a polynomial-time transformation from an arbitrary instance f of the SAT problem to a cast puzzle c 2 such that f is satisfiable if and only if c 2 is solvable.

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“…Even with such approximations the problem of finding strategies to maximize the score is NP-Complete [4]. Furthermore, is has been shown in [6] that the game cannot last forever and that it finishes with probability 1.…”

Section: Introductionmentioning

confidence: 99%

On the evolution of artificial Tetris players

Boumaza

2009

2009 IEEE Symposium on Computational Intelligence and Games

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In the paper, we focus the use of evolutionary algorithms to learn strategies to play the game of Tetris. We describe the problem and discuss the nature of the search space. We present experiments to illustrate the learning process of our artificial player, and provide a new procedure to speed up the learning time. The results we present compare with the best known artificial player, and show how our evolutionary algorithm is able to rediscover player strategies previously published. Finally we provide some ideas to improve the performance of artificial Tetris players.

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Tetris is Hard, Even to Approximate

Cited by 76 publications

References 7 publications

Graph kernels and Gaussian processes for relational reinforcement learning

Graph kernels and Gaussian processes for relational reinforcement learning

NP-Hard and k-EXPSPACE-Hard Cast Puzzles

On the evolution of artificial Tetris players

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