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Tetris is Hard, Even to Approximate
[article]

2002
*
arXiv
*
pre-print

We prove that in the offline version of

arXiv:cs/0210020v1
fatcat:wmz5epkx5veyvhdr7sjwus4u3q
*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] . ...##
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TETRIS IS HARD, EVEN TO APPROXIMATE

2004
*
International journal of computational geometry and applications
*

We have shown that it

doi:10.1142/s0218195904001354
fatcat:lf4si2gitnbd3kwwxxbvcht4fe
*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 ...##
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Tetris is Hard, Even to Approximate
[chapter]

2003
*
Lecture Notes in Computer Science
*

We have shown that it

doi:10.1007/3-540-45071-8_36
fatcat:62e2qybfafbtjmq7oytzs5fuha
*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 ...##
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High-Dimensional Function Approximation for Knowledge-Free Reinforcement Learning

2015
*
Proceedings of the 2015 on Genetic and Evolutionary Computation Conference - GECCO '15
*

None of them really work on (SZ-)

doi:10.1145/2739480.2754783
dblp:conf/gecco/JaskowskiSLK15
fatcat:b26etv63nnccjirkvw7a6wqiz4
*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 ...##
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Learning Tetris Using the Noisy Cross-Entropy Method

2006
*
Neural Computation
*

The cross-entropy method

doi:10.1162/neco.2006.18.12.2936
pmid:17052153
fatcat:ivlucsytgjcyfk4kf7d3p5pufq
*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. ...##
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How Fast Can We Play Tetris Greedily With Rectangular Pieces?
[article]

2022
*
arXiv
*
pre-print

On the other hand, we show that there

arXiv:2202.10771v2
fatcat:npv4akt6izbnvecyfqf35qkmyy
*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. ...##
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Box Covers and Domain Orderings for Beyond Worst-Case Join Processing
[article]

2021
*
arXiv
*
pre-print

Our third contribution

arXiv:1909.12102v3
fatcat:o3ukbs5i45aprfs3kwpxgyq7em
*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. ...##
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Building Controllers for Tetris

2009
*
ICGA Journal
*

An immediate interest of this review

doi:10.3233/icg-2009-32102
fatcat:usqox5shwrduvnx4h5s5gieebi
*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. ...##
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Tetris: A Study of Randomized Constraint Sampling
[chapter]

2006
*
Probabilistic and Randomized Methods for Design under Uncertainty
*

Acknowledgments This material

doi:10.1007/1-84628-095-8_6
fatcat:qwmq57dlq5enveio6b3teymeku
*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. ...##
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Multi-resource packing for cluster schedulers

2014
*
Proceedings of the 2014 ACM conference on SIGCOMM - SIGCOMM '14
*

We present

doi:10.1145/2619239.2626334
dblp:conf/sigcomm/GrandlAKRA14
fatcat:qulrhczvufggrhgrcjm65qibem
*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 ...##
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Multi-resource packing for cluster schedulers

2014
*
Computer communication review
*

We present

doi:10.1145/2740070.2626334
fatcat:og5n7krtojea7eb67zofmxgpv4
*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 ...##
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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
*

Part of what makes

doi:10.1145/3071178.3071195
dblp:conf/gecco/GillespieGS17
fatcat:fp5fjjtjs5cwzixilygjwj67ym
*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] . ...##
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Bust-a-Move/Puzzle Bobble Is NP-complete
[chapter]

2016
*
Lecture Notes in Computer Science
*

Our proof applies

doi:10.1007/978-3-319-48532-4_9
fatcat:37jdrvyom5c33cuq2emgcbqkey
*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. ...##
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How to Count Triangles, without Seeing the Whole Graph
[article]

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. ...

##
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Integrating Guidance into Relational Reinforcement Learning

2004
*
Machine Learning
*

The first problem

doi:10.1023/b:mach.0000039779.47329.3a
fatcat:63rokvu2knehjc5q7j3m2rxuae
*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. ...
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