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Tetris is Hard, Even to Approximate
 COCOON
, 2003
"... In the popular computer game of Tetris, the player is given a sequence of tetromino pieces and must pack them into a rectangular gameboard initially occupied by a given configuration of filled squares; any completely filled row of the gameboard is cleared and all pieces above it drop by one row. ..."
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Cited by 45 (2 self)
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In the popular computer game of Tetris, the player is given a sequence of tetromino pieces and must pack them into a rectangular gameboard initially occupied by a given configuration of filled squares; any completely filled row of the gameboard is cleared and all pieces above it drop by one row. We prove that in the o#ine version of Tetris, it is NPcomplete to maximize the number of cleared rows, maximize the number of tetrises (quadruples of rows simultaneously filled and cleared), minimize the maximum height of an occupied square, or maximize the number of pieces placed before the game ends. We furthermore show the extreme inapproximability of the first and last of these objectives to within a factor of p , when given a sequence of p pieces, and the inapproximability of the third objective to within a factor of 2#, for any # > 0. Our results
How to Lose at Tetris
 Mathematical Gazette
, 1997
"... This paper addresses the question: "can you `win' the game Tetris?" Designed by Soviet mathematician Alexey Pazhitnov in the late eighties and imported to the United States by Spectrum Holobyte, Tetris won a record number of software awards in 1989 [4]. Versions of Tetris are sold for ..."
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Cited by 27 (0 self)
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This paper addresses the question: "can you `win' the game Tetris?" Designed by Soviet mathematician Alexey Pazhitnov in the late eighties and imported to the United States by Spectrum Holobyte, Tetris won a record number of software awards in 1989 [4]. Versions of Tetris are sold for most personal computers. There are Tetris arcade games, Tetris Nintendo cartridges, and handheld Tetris games; Tetris has been played on machines ranging from mainframes to calculators. The game's success has prompted the invention of several similar games, including Hextris, Welltris, and Wordtris. Although mathematicians have spent many hours "studying" Tetris, surprisingly little is known about the mathematical properties of the game. Much research has been done on the subject of covering rectangles with sets of polyominoes [2,3,5,6]; Tetris adds a new twist to this familiar problem. The game takes place on a grid or "board" ten units wide and twenty units tall. When the game starts, the board is empty. Then tetrominoes, groups of four connected "cells", each cell covering exactly one grid square, appear at the top of the board and fall row by row toward the bottom of the board (see Figure 1). When a tetromino reaches the bottom or a point where it can fall no further without two or more cells overlapping, it remains in that spot and another tetromino (randomly selected from the set of seven possible tetrominoes) appears at the top of the board. The player uses rotations and horizontal translations to orient the tetrominoes as they fall, attempting to cover rows of the board with cells. When a row is covered, the cells on that row are removed from the board and the cells of the rows above drop down to fill the Figure 1: The Tetrominoes Figure 2: ZTetrominoes gap. Figure 1 shows how ...
TETRIS IS HARD, EVEN TO APPROXIMATE
, 2003
"... In the popular computer game of Tetris, the player is given a sequence of tetromino pieces and must pack them into a rectangular gameboard initially occupied by a given configuration of filled squares; any completely filled row of the gameboard is cleared and all filled squares above it drop by one ..."
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Cited by 16 (1 self)
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In the popular computer game of Tetris, the player is given a sequence of tetromino pieces and must pack them into a rectangular gameboard initially occupied by a given configuration of filled squares; any completely filled row of the gameboard is cleared and all filled squares above it drop by one row. We prove that in the offline version of Tetris, it is NPcomplete to maximize the number of cleared rows, maximize the number of tetrises (quadruples of rows simultaneously filled and cleared), minimize the maximum height of an occupied square, or maximize the number of pieces placed before
Approximate Dynamic Programming via a Smoothed Linear Program
"... We present a novel linear program for the approximation of the dynamic programming costtogo function in highdimensional stochastic control problems. LP approaches to approximate DP have typically relied on a natural ‘projection ’ of a well studied linear program for exact dynamic programming. Such ..."
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Cited by 12 (2 self)
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We present a novel linear program for the approximation of the dynamic programming costtogo function in highdimensional stochastic control problems. LP approaches to approximate DP have typically relied on a natural ‘projection ’ of a well studied linear program for exact dynamic programming. Such programs restrict attention to approximations that are lower bounds to the optimal costtogo function. Our program—the ‘smoothed approximate linear program’— is distinct from such approaches and relaxes the restriction to lower bounding approximations in an appropriate fashion while remaining computationally tractable. Doing so appears to have several advantages: First, we demonstrate substantially superior bounds on the quality of approximation to the optimal costtogo function afforded by our approach. Second, experiments with our approach on a challenging problem (the game of Tetris) show that the approach outperforms the existing LP approach (which has previously been shown to be competitive with several ADP algorithms) by an order of magnitude. 1.
The Smoothed Approximate Linear Program
, 2009
"... We present a novel linear program for the approximation of the dynamic programming costtogo function in highdimensional stochastic control problems. LP approaches to approximate DP have typically relied on a natural ‘projection ’ of a well studied linear program for exact dynamic programming. Such ..."
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Cited by 10 (0 self)
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We present a novel linear program for the approximation of the dynamic programming costtogo function in highdimensional stochastic control problems. LP approaches to approximate DP have typically relied on a natural ‘projection ’ of a well studied linear program for exact dynamic programming. Such programs restrict attention to approximations that are lower bounds to the optimal costtogo function. Our program—the ‘smoothed approximate linear program’— is distinct from such approaches and relaxes the restriction to lower bounding approximations in an appropriate fashion while remaining computationally tractable. Doing so appears to have several advantages: First, we demonstrate substantially superior bounds on the quality of approximation to the optimal costtogo function afforded by our approach. Second, experiments with our approach on a challenging problem (the game of Tetris) show that the approach outperforms the existing LP approach (which has previously been shown to be competitive with several ADP algorithms) by an order of magnitude. 1.
Applying reinforcement learning to Tetris
, 2005
"... This paper investigates the possible application of reinforcement learning to Tetris. The author investigates the background of Tetris, and qualifies it in a mathematical context. The author discusses reinforcement learning, and considers historically successful applications of it. Finally the aut ..."
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This paper investigates the possible application of reinforcement learning to Tetris. The author investigates the background of Tetris, and qualifies it in a mathematical context. The author discusses reinforcement learning, and considers historically successful applications of it. Finally the author discusses considerations surrounding implementation. 1
Apply Ant Colony Optimization to Tetris
"... Tetris is a falling block game where the player’s objective is to arrange a sequence of different shaped tetrominoes smoothly in order to survive. In the intelligence games, agent imitates the real player and chooses the best move based on a linear value function. In this paper, we apply Ant Colony ..."
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Tetris is a falling block game where the player’s objective is to arrange a sequence of different shaped tetrominoes smoothly in order to survive. In the intelligence games, agent imitates the real player and chooses the best move based on a linear value function. In this paper, we apply Ant Colony Optimization (ACO) method to learn the weights of the function, trying to search an optimal weightpath in the weight graph. We use dynamic heuristic to prevent premature convergence to local optima. Our experimental result is better than most of traditional reinforcement learning methods.
Abstract Tetris and Decidability ⋆
"... We consider a variant of Tetris where the sequence of pieces (together with their orientation and horizontal position, which cannot be changed anymore) is generated by a finite state automaton. We show that it is undecidable, given such an automaton, and starting from an empty game board, whether on ..."
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We consider a variant of Tetris where the sequence of pieces (together with their orientation and horizontal position, which cannot be changed anymore) is generated by a finite state automaton. We show that it is undecidable, given such an automaton, and starting from an empty game board, whether one of the generated sequences leaves an empty game board. This is contrasted with more common situations where piece translations and rotations are allowed.
Using a Genetic Algorithm to Weight an Evaluation Function for Tetris
"... Tetris is a popular videogame invented by Alexey Pajitnov. An agent that plays Tetris must be able to place pieces in good positions without knowledge of what pieces will follow. One way such an agent can work is to use an evaluation function to place the current piece. This evaluation function is ..."
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Tetris is a popular videogame invented by Alexey Pajitnov. An agent that plays Tetris must be able to place pieces in good positions without knowledge of what pieces will follow. One way such an agent can work is to use an evaluation function to place the current piece. This evaluation function is a weighted sum of features from the board. We used a genetic algorithm, based on Genitor, to discover these weights. We tried several things to improve the efficiency of the search for the weights, including different fitness evaluations and crossover operations.
Adapting Reinforcement Learning to Tetris
, 2005
"... This paper discusses the application of reinforcement learning to Tetris. Tetris and reinforcement learning are both introduced and defined, and relevent research is discussed. An agent based on existing research is implemented and investigated. A reduced representation of the Tetris state space is ..."
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This paper discusses the application of reinforcement learning to Tetris. Tetris and reinforcement learning are both introduced and defined, and relevent research is discussed. An agent based on existing research is implemented and investigated. A reduced representation of the Tetris state space is then developed, and several new agents are implemented around this state space. The implemented agents all display successful learning, and show proficiency within certain conditions.