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Tetris is Hard, Even to Approximate [article]

Erik D. Demaine, Susan Hohenberger, David Liben-Nowell
2002 arXiv   pre-print
We prove that in the offline version of Tetris, it is NP-complete to maximize the number of cleared rows, maximize the number of tetrises (quadruples of rows simultaneously filled and cleared), minimize  ...  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;  ...  Thanks also to Josh Tauber for pointing out the puzzles in Games Magazine [7] .  ... 
arXiv:cs/0210020v1 fatcat:wmz5epkx5veyvhdr7sjwus4u3q

TETRIS IS HARD, EVEN TO APPROXIMATE

RON BREUKELAAR, ERIK D. DEMAINE, SUSAN HOHENBERGER, HENDRIK JAN HOOGEBOOM, WALTER A. KOSTERS, DAVID LIBEN-NOWELL
2004 International journal of computational geometry and applications  
We have shown that it is NP-hard to optimize-or even approximate-a number of natural objectives for offline Tetris. 1) ?  ...  It would also be interesting to analyze theoretically the online version of Tetris, perhaps considering a probabilistically generated piece sequence.  ...  We also prove the extreme inapproximability of the most natural objective functions: for any constant ε > 0, it is NP-hard to approximate to within a factor of p 1−ε the maximum number of pieces that can  ... 
doi:10.1142/s0218195904001354 fatcat:lf4si2gitnbd3kwwxxbvcht4fe

Tetris is Hard, Even to Approximate [chapter]

Erik D. Demaine, Susan Hohenberger, David Liben-Nowell
2003 Lecture Notes in Computer Science  
We have shown that it is NP-hard to optimize-or even approximate-a number of natural objectives for offline Tetris. 1) ?  ...  It would also be interesting to analyze theoretically the online version of Tetris, perhaps considering a probabilistically generated piece sequence.  ...  We also prove the extreme inapproximability of the most natural objective functions: for any constant ε > 0, it is NP-hard to approximate to within a factor of p 1−ε the maximum number of pieces that can  ... 
doi:10.1007/3-540-45071-8_36 fatcat:62e2qybfafbtjmq7oytzs5fuha

High-Dimensional Function Approximation for Knowledge-Free Reinforcement Learning

Wojciech Jaśkowski, Marcin Szubert, Paweł Liskowski, Krzysztof Krawiec
2015 Proceedings of the 2015 on Genetic and Evolutionary Computation Conference - GECCO '15  
None of them really work on (SZ-)Tetris, they do not even come close to the performance of the evolutionary approaches. 1 Not easy for direct search methods State-Evaluation Function and Action Selection  ...  Hard for value function-based methods There are many RL algorithms for approximating the value functions.  ...  Summary RL Perspective 1 High-dimensional representation (systematic n-tuple network) to: Make TD work at all on this problem The best player to date 2 (nearly 300 lines on average) > hand-coded strategy  ... 
doi:10.1145/2739480.2754783 dblp:conf/gecco/JaskowskiSLK15 fatcat:b26etv63nnccjirkvw7a6wqiz4

Learning Tetris Using the Noisy Cross-Entropy Method

István Szita, András Lörincz
2006 Neural Computation  
The cross-entropy method is an efficient and general optimization algorithm.  ...  However, its applicability in reinforcement learning (RL) seems to be limited because it often converges to suboptimal policies.  ...  Finding the optimal strategy is NP-hard even if the sequence of tetrominoes is known in advance.  ... 
doi:10.1162/neco.2006.18.12.2936 pmid:17052153 fatcat:ivlucsytgjcyfk4kf7d3p5pufq

How Fast Can We Play Tetris Greedily With Rectangular Pieces? [article]

Justin Dallant, John Iacono
2022 arXiv   pre-print
On the other hand, we show that there is a data structure supporting both operations in O(n^1/2log^3/2n) time on boards of width n^O(1), matching the lower bound up to a n^o(1) factor.  ...  them drop, and a row does not disappear once it is full.  ...  Tetris is hard, even to approximate. In Tandy J.  ... 
arXiv:2202.10771v2 fatcat:npv4akt6izbnvecyfqf35qkmyy

Box Covers and Domain Orderings for Beyond Worst-Case Join Processing [article]

Kaleb Alway, Eric Blais, Semih Salihoglu
2021 arXiv   pre-print
Our third contribution is a Õ(N)-time approximation algorithm called ADORA to compute domain orderings, under which one can compute a box cover of size Õ(K^r), where K is the minimum box cover for Q under  ...  Second, we show that finding a domain ordering to minimize the box cover size and certificate is NP-hard through a reduction from the 2 consecutive block minimization problem on boolean matrices.  ...  The problem is known to be NP-complete, even when the polygon is hole-free [9] and MaxSNP-hard for polygons with holes [5] . There are several approximation algorithms for the problem.  ... 
arXiv:1909.12102v3 fatcat:o3ukbs5i45aprfs3kwpxgyq7em

Building Controllers for Tetris

Christophe Thiery, Bruno Scherrer
2009 ICGA Journal  
An immediate interest of this review is illustrated. Straightforwardly gathering ideas from different works may lead to new ideas.  ...  Finally, we briefly discuss how this implementation allowed us to win the Tetris-domain prize of the 2008 Reinforcement Learning Competition.  ...  Such a problem is known to be computationally hard to solve.  ... 
doi:10.3233/icg-2009-32102 fatcat:usqox5shwrduvnx4h5s5gieebi

Tetris: A Study of Randomized Constraint Sampling [chapter]

Vivek F. Farias, Benjamin Van Roy
2006 Probabilistic and Randomized Methods for Design under Uncertainty  
Acknowledgments This material is based upon work supported by the National Science Foundation under Grant ECS-9985229 and by the Office of Naval Research under Grant MURI N00014-00-1-0637.  ...  is NPcomplete, even to approximate.  ...  even infinity.  ... 
doi:10.1007/1-84628-095-8_6 fatcat:qwmq57dlq5enveio6b3teymeku

Multi-resource packing for cluster schedulers

Robert Grandl, Ganesh Ananthanarayanan, Srikanth Kandula, Sriram Rao, Aditya Akella
2014 Proceedings of the 2014 ACM conference on SIGCOMM - SIGCOMM '14  
We present Tetris, a multi-resource cluster scheduler that packs tasks to machines based on their requirements of all resource types.  ...  needs change with time and with the machine that the task is placed at.  ...  APX-Hard means that there is no asymptotic polynomial time approximation unless P=NP; it is a strict subset of NP-hard. a network-intensive task has a higher dot product. is heuristic prevents over-allocation  ... 
doi:10.1145/2619239.2626334 dblp:conf/sigcomm/GrandlAKRA14 fatcat:qulrhczvufggrhgrcjm65qibem

Multi-resource packing for cluster schedulers

Robert Grandl, Ganesh Ananthanarayanan, Srikanth Kandula, Sriram Rao, Aditya Akella
2014 Computer communication review  
We present Tetris, a multi-resource cluster scheduler that packs tasks to machines based on their requirements of all resource types.  ...  needs change with time and with the machine that the task is placed at.  ...  APX-Hard means that there is no asymptotic polynomial time approximation unless P=NP; it is a strict subset of NP-hard. a network-intensive task has a higher dot product. is heuristic prevents over-allocation  ... 
doi:10.1145/2740070.2626334 fatcat:og5n7krtojea7eb67zofmxgpv4

Comparing direct and indirect encodings using both raw and hand-designed features in tetris

Lauren E. Gillespie, Gabriela R. Gonzalez, Jacob Schrum
2017 Proceedings of the Genetic and Evolutionary Computation Conference on - GECCO '17  
Part of what makes Tetris so hard and therefore so interesting is the wide variety of pieces and how it is impossible to place every sequence of blocks in a way that the player will not lose, as research  ...  Furthermore, Tetris is an NPcomplete problem, even when the player knows the identity and order of all the pieces [5] .  ... 
doi:10.1145/3071178.3071195 dblp:conf/gecco/GillespieGS17 fatcat:fp5fjjtjs5cwzixilygjwj67ym

Bust-a-Move/Puzzle Bobble Is NP-complete [chapter]

Erik D. Demaine, Stefan Langerman
2016 Lecture Notes in Computer Science  
Our proof applies to the perfectinformation version where the bubble sequence is known in advance, and uses just three colors.  ...  We prove that the classic 1994 Taito video game, known as Puzzle Bobble or Bust-a-Move, is NPcomplete.  ...  Thus, even approximating the maximum number of poppable bubbles better than a factor of n 1−ε is NP-hard (as in Tetris [1] ). See the full paper [2] for omitted details.  ... 
doi:10.1007/978-3-319-48532-4_9 fatcat:37jdrvyom5c33cuq2emgcbqkey

How to Count Triangles, without Seeing the Whole Graph [article]

Suman K. Bera, C. Seshadhri
2020 arXiv   pre-print
Even sampling a uniform random vertex is a hard task in this model. Despite these challenges, we design a provable and practical algorithm, TETRIS, for triangle counting in this model.  ...  TETRIS is the first provably sublinear algorithm (for most natural parameter settings) that approximates the triangle count in the random walk model, for graphs with low mixing time.  ...  ACKNOWLEDGMENTS The authors would like to thank the anonymous reviewers for their valuable feedback.  ... 
arXiv:2006.11947v1 fatcat:54qcbkdt5rh2npk72wb27cn35i

Integrating Guidance into Relational Reinforcement Learning

Kurt Driessens, Sašo Džeroski
2004 Machine Learning  
The first problem is often solved by learning a generalization of the encountered examples (e.g., using a neural net or decision tree).  ...  Different types of policies and different strategies to supply guidance through these policies are discussed and evaluated experimentally in several relational domains to show the merits of the approach  ...  The Tetris game It is quite hard (if not impossible) to generate an optimal or even "human level" strategy for the Tetris game.  ... 
doi:10.1023/b:mach.0000039779.47329.3a fatcat:63rokvu2knehjc5q7j3m2rxuae
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