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Tetris is Hard, Even to Approximate
[article]
2002
arXiv
pre-print
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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
2004
International journal of computational geometry and applications
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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]
2003
Lecture Notes in Computer Science
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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
2015
Proceedings of the 2015 on Genetic and Evolutionary Computation Conference - GECCO '15
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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
2006
Neural Computation
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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]
2022
arXiv
pre-print
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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]
2021
arXiv
pre-print
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2020 pre-print arXiv:1909.12102v2 fatcat:imwn3eztbzbrdhnkuvdijeywjuOther Versions
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
2009
ICGA Journal
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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]
2006
Probabilistic and Randomized Methods for Design under Uncertainty
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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
2014
Proceedings of the 2014 ACM conference on SIGCOMM - SIGCOMM '14
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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
2014
Computer communication review
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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
2017
Proceedings of the Genetic and Evolutionary Computation Conference on - GECCO '17
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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]
2016
Lecture Notes in Computer Science
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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]
2020
arXiv
pre-print
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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
2004
Machine Learning
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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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