Approximately Optimal Search on a Higher-dimensional Sliding Puzzle

\'Erika Rold\'an; Miguel O'Malley; Nono SC Merleau; Sayan Mukherjee

REVIEW 4 major objections 5 minor 1 cited by

Approximately Optimal Search on a Higher-dimensional Sliding Puzzle

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read On the paper's own account, exact A* search solves dimension-3 and most dimension-4 hypercube sliding puzzles, while evolutionary and reinforcement-learning searches still solve dimension-5 instances with median move counts usually below…

desk verdict A first benchmark of search methods on the hypercube k-rule sliding puzzle, but the preprint's own equations, pseudocode, and tables contradict the reported results. read the letter →

arxiv 2412.01937 v1 pith:I57DUTOD submitted 2024-12-02 cs.AI cs.DMcs.LGcs.NE

Nono SC Merleau , Miguel O'Malley , Érika Roldán , Sayan Mukherjee This is my paper · ORCID

classification cs.AIcs.DMcs.LGcs.NE

keywords higher-dimensionalslidingpuzzlehypercubek-ruleA*searchevolutionaryalgorithmreinforcementlearningdiameterNP-hard

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies generalized sliding puzzles on a d-dimensional hypercube: colored rings occupy distinct vertices and may slide only across a k-dimensional face whose other vertices are empty. Its central claim is a computational frontier: in dimension three all three tested methods—A* search, an evolutionary algorithm, and reinforcement learning—solve the puzzles, and A* certifies optimal solutions; in dimensions four and five, exact A* search becomes infeasible for the harder levels, while the two stochastic methods still produce short, 'generally acceptable' solutions, with medians usually below thirty moves. The authors present this benchmark as evidence that approximately optimal search can cover a regime that exact search cannot reach for this NP-hard puzzle family. If the claim is right, the practical conclusion is that the high-dimensional cases are algorithmically tractable in an approximate sense, not that they become easy.

What carries the argument

The load-bearing object is the (d,k,l)-cubical sliding puzzle on the hypercube $Q_d$: $2^d-l$ colored rings occupy distinct vertices, and a k-move slides a ring to any vertex of a k-dimensional face that contains no other ring. The A* implementation depends on the admissible lower-bound heuristic $h_T(C)=\sum_i \lceil |C_i-T_i|/k\rceil$, whose monotonicity is used to guarantee optimality. The EA carries the argument through a fitness $f_a(C,T)=1/(1+L-h(C,T))$ and a selection force that linearly combines that fitness with the inverse move count. The RL method carries its share through a reward function seeded by a breadth-first expansion of the target configuration, so that early random walks are biased toward configurations with known short paths to the goal.

What would settle it

Run the released EA with the objective exactly as printed in Eqs. (3)-(4): if h(C,T) is the number of mismatched ring-target pairs, then $f_a(C,T)=1/(1+L-h)$ reaches its maximum 1 at h=L, i.e. at a configuration where every ring is in the wrong place, so the loop 'while $f_a \neq 1$' should never stop for any nontrivial puzzle. If the code still solves puzzles, the experiments are not testing the algorithm described; if it does not, the reported EA successes would not be reproducible.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the difficulty wall for exact search is not a wall for stochastic search. For d=3, A* yields provably optimal move counts between 4 and 10, and both RL and EA match or approach these optima. For d=4, A* still solves levels 0-3 optimally but fails on level 4 within the time limit; for d=5, A* finds no solution, while EA solves the k=1, k=2, and k=3 puzzles with median move counts of 12, 9, and 7, and RL solves them with medians of 8, 5, and 4. The paper's own tables record one sharp exception to the 'median below 30' slogan: for d=5 and k=4, RL has median 161 moves and EA never terminates successfully. The authors infer that EA is consistently the fastest method and RL the most accurate stochastic method, with A* remaining the method of choice whenever optimality is required and the dimension is small.

Load-bearing premise

The EA results presuppose that the implemented fitness function rewards rings that already match their target colours, but the printed equations define h as a mismatch count, so the printed fitness is maximized when zero rings match and the stopping condition $f_a=1$ can never be reached for $L>0$.

Editorial extensions

If this is right

A* search certifies optimal solutions for d=3 and for most d=4 puzzles, with optimum move counts between 4 and 10, so for small dimensions exact search remains the reference method.
For d=5, A* times out, but EA solves k=1,2,3 puzzles with median moves 12, 9, and 7, and RL solves the same cases with medians 8, 5, and 4; approximate solutions remain available where optimal ones are not.
EA is consistently the fastest in CPU time but its success rate collapses for d=4, k=3 to under 5 percent, whereas RL keeps near-100 percent success across tested levels and face dimensions.
The A* branching factor on these puzzles is at least $2^k-1$, so exact search cost grows at least like $(2^k-1)^n$ in solution length $n$; this is the concrete combinatorial wall that the stochastic methods are claimed to bypass.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

[Editorial inference] If the printed fitness in Eqs. (3)-(4) is taken literally—h counts mismatches—the EA as described cannot terminate, so the reported EA numbers must come from an unstated corrected objective; re-running with the stated equations would be a direct test of which algorithm the benchmark actually evaluates.
[Editorial inference] The target-seeded breadth-first weighting used for RL could be fused with A* as a bidirectional or windowed search, potentially pushing exact solutions into some d=5 instances where the current A* stalls.
[Editorial inference] The paper compares medians and CPU times, but the full distributions suggest a three-way trade-off among accuracy, variance, and speed; a Pareto-style summary would let practitioners choose a method by difficulty level rather than by dimension alone.
[Editorial inference] Because the difficulty levels appear to have been generated only for k=d-1, the reported monotone increase of move count with level and k may be an artifact of level generation; generating levels independently per k could separate puzzle hardness from algorithm behavior.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short take: the benchmark idea is legitimate and new—nobody has run A*, EA, and RL on this [4] puzzle family before—and the comparison across d=3,4,5 with code on GitHub is potentially useful. The qualitative story (A* optimal but fails at scale; EA fast but poor at minimizing; RL slower but more accurate) is plausible. But as written the paper doesn't support its own conclusions.

First, the EA objective is inverted. Eq. (3)-(4) define h as a count of mismatches and then set fa=1/(1+L-h). That function is maximized when h=L, i.e. when nothing matches, and fa=1 is unreachable for L>0. If the code follows the equations, the stopping condition can never fire; if it follows a corrected objective, then the experiments don't test the algorithm the paper describes.

Second, Algorithm 4 line 17 is not executable: 'while C != T and c < 0 and s(C) is not undefined'—c starts at 0, so the loop is never entered. The surrounding pseudocode also uses undefined functions. That makes the RL results unreproducible from the text.

Third, the abstract's headline claim—RL and EA provide solutions with median under 30 when A* fails—is directly contradicted by Table 5: d=5, k=4, RL success 100% but median 161. And Table 4 shows EA success 0% for that same puzzle, so the plural claim fails too.

Fourth, the A* monotonicity argument in Section 3.1 is not a proof; saying 'since d(C,C') is precisely the number of moves' just restates the question. The heuristic is not shown to be monotone, so the optimality guarantee is unsupported.

Fifth, Tables 4 and 5 contain identical RL and EA rows for d=4, k=1 and k=2 at every level. That suggests a data-handling or table-generation error. The GitHub repository might clear this up, but the preprint alone doesn't.

I don't see derivation-based circularity and the self-citation to [4] is appropriate. But these are load-bearing mechanical errors. I'd not desk-reject the subject matter: the puzzle benchmark could be a useful reference. But this version should not go to referees. I'd send it back to the authors with instructions to fix the objective, the pseudocode, the summary stats, and the table duplication, then resubmit.

Referee Report

4 major / 5 minor

Summary. The paper studies a higher-dimensional sliding puzzle on the d-dimensional hypercube with the k-rule move constraint. It proposes three search methods—A*, an evolutionary algorithm (EA), and a reinforcement learning (RL) agent—and benchmarks them on puzzles of dimensions d=3, 4, and 5 across several difficulty levels. The main reported conclusions are that A* solves the d=3 and most d=4 puzzles optimally, while for larger dimensions EA and RL still return approximate solutions, with the EA being fastest and RL being more accurate. The paper also provides a CPU-time comparison and claims that the methods go beyond the theoretical reachability results of prior work.

Significance. If the claims held, the paper would provide a useful practical benchmark showing that stochastic search can handle hypercube sliding puzzles beyond the reach of exact search. The authors ship a public code repository, report experiments over multiple dimensions, face dimensions, and difficulty levels, and provide CPU-time comparisons; these are genuine strengths. However, the central empirical and algorithmic claims are not currently supported: the EA objective function as written is inconsistent with the reported stopping behavior, the RL pseudocode is not executable as printed, and the paper's headline claim about median moves under 30 is contradicted by its own Table 5. Because these issues affect the core results, the contribution cannot be accepted in its present form.

major comments (4)

[§3.3, Eq. (3)-(4), Algorithm 2] The fitness function in Eq. (3)-(4) is internally inconsistent with the stated stopping criterion. h(C,T) is defined as the number of mismatches (δ=1 when C_i≠T_i), so L−h is the number of matches. The fitness fa=1/(1+L−h) is therefore maximized when h=0, but its maximum value is 1/(1+L), not 1. Conversely, fa=1 only when h=L, i.e., when every ring is mismatched. This directly contradicts Algorithm 2 line 4, which stops when fa(C_b,T)=1, and Appendix A, which equates the maximum fitness with hamming distance 0. The EA results in Table 4 therefore cannot be produced by the algorithm as described unless the implemented objective differs from the printed one; the equations and the experimental code must be reconciled and the experiments rerun.
[Appendix D, Algorithm 4] The RL pseudocode is not executable as written. Line 4 tests `n(T) < N`, but `n` is never defined and `N` is not an input parameter (the input is the branching parameter `P`). Line 17 contains the loop condition `c < 0`, which is never true because c starts at 0 and is only incremented; the surrounding text in §3.2 says episodes terminate when more steps have been taken than the best known path, which would require a condition such as `c < o` rather than `c < 0`. As printed, the algorithm either loops forever or terminates immediately, so the RL results in Table 5 are not reproducible from the stated method.
[Abstract vs. Tables 4 and 5] The abstract's central claim that, when the dimension increases, "RL and EA methods can still provide a generally acceptable solution, i.e. a distribution of a number of moves with a median value of less than 30" is directly contradicted by the paper's own data. Appendix F, Table 5, for d=5, l=26, k=4, level 0, reports RL success 100% with min 62, max 316, and median 161 moves. Table 4 reports an EA success rate of 0% for the same puzzle. Thus the reported median is more than five times the advertised bound, and the plural "RL and EA methods" is not supported for this case. The headline claim must be corrected or substantially narrowed.
[Tables 4 and 5] For d=4, k=1 and k=2, the EA performance table (Table 4) and the RL performance table (Table 5) report identical min, max, and median values for every difficulty level, including entries such as level 1: min 8, max 60, median 14, and level 4: min 12, max 28, median 16. Since these are independent stochastic algorithms run over 150 trials, identical summary statistics across all levels are not plausible unless the same data were used for both tables. This data-consistency problem must be resolved before the EA-vs-RL comparison in §4.2 can be trusted.

minor comments (5)

[Algorithm 2 comments] The comments in Algorithm 2 refer to "the length of the target RNA structure," "base-pair distance," and a Lévy or Binomial distribution, which appear to be copied from an RNA-folding EA. These should be replaced with puzzle-specific terminology.
[§4.3, Proposition 1] The proof states the complexity is "at least O((2^k−1)^n)", but the following sentence says "the complexity of A* search for a face dimension k=4 is at most O(7^n)"; the latter should be "at least" or "of order" to be consistent.
[Figure 3 caption] The caption of Figure 3 says the puzzle parameters are k=2, d=3, l=4, but panel (b) shows d=4; the caption should describe both panels accurately.
[§4.2] The sentence beginning "Among the three algorithms ... EA and RL techniques are ˚a both choose" contains a typographical error that obscures the intended meaning.
[Appendix A] The stopping criteria in Appendix A state that the hamming distance of the best agent to the target is 1, whereas the main text (§3.3) states the stopping condition is hamming distance 0; these should be made consistent.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper is an empirical benchmark whose A*, RL, and EA results are independent of the algorithms' own outputs, and the only self-citation is not load-bearing.

full rationale

The paper is an empirical benchmark, not a derivation, and its central claims (A* optimal on d=3 and most d=4; EA/RL solving d=5 where A* times out) are supported by Tables 2, 4, and 5, which are independent of the algorithms' own outputs. No fitted parameter is relabeled as a prediction: EA hyperparameters (c*=1.8, alpha=0.2) are tuned on one d=3 instance in Appendix A and then fixed for the other puzzles, and RL hyperparameters are fixed as described in Section 3.2. The only self-citation, [4], is used in Proposition 1 to bound A*'s branching factor; that prior theorem is a mathematical fact about the puzzle model and is not used to define the experimental results, and the A* failure claim is established empirically in Table 2 rather than by the proposition. There is an internal inconsistency in the EA fitness function as printed: Eq. (3)-(4) define h as a mismatch count, so fa is maximized when rings do not match and the stopping condition fa=1 is unreachable; this is a correctness/bug concern, not a circular reduction, because no quantity is fit to its own output. The abstract's 'median less than 30' claim is also not supported by Table 5 for d=5, k=4 (median 161), but a data contradiction is distinct from circularity. No step in the paper equates a predicted quantity to its input by construction.

Assumptions & free parameters 6 free parameters · 3 assumptions · 0 invented entities

No new particles, mediators, forces, dimensions, or conserved quantities are introduced. The k-rule puzzle and its terminology come from prior work [4]. The ledger instead records hand-tuned algorithm hyperparameters that the reported performance depends on, plus background mathematical assumptions from prior literature.

free parameters (6)

EA mutation Zipf exponent c = c* = 1.8
Tuned on d=3, k=2, level 0 (Appendix A, Figure 8) among values 1..7, then used for all reported EA runs.
EA selection force alpha = 0.2
Chosen in parameter analysis (Section 4.1); balances distance-to-target and move count.
EA population size N and max generations T = N = 1000, T = 1000
Algorithm settings for all reported runs; no sensitivity analysis beyond c and alpha.
RL branching states P = 1000 (most puzzles), 100000 (d=5)
Chosen by hand based on configuration-space size; authors state optimal P left to future work.
RL learning rate alpha and discount gamma = alpha = 0.05, gamma = 0.95
Fixed for all experiments without a tuning study.
RL episodes and trials = 1000 iterations, 150 runs per puzzle
Computational budget rather than a physical model; results depend on it.

assumptions (3)

domain assumption Background solvability, parity, and branching-factor properties of cubical puzzles from Beyer et al. [4] are correct and apply to the random starting configurations used.
The paper identifies an impossible difficulty level and uses the 2^k-1 branching lower bound in Proposition 1 based on [4] without re-deriving them.
standard math The A* evaluation heuristic defined in Definition 2 is admissible and monotonic.
Optimality of A* is invoked from [6]; the paper asserts monotonicity without a valid proof, and the assertion is false for k>1 because one move can reduce the heuristic by up to k.
domain assumption Each ring's distance can be summed independently as a lower bound on total moves.
Used in Definition 2; it is a lower bound, but ignores all blocking interactions and thus may be weak, affecting A* efficiency but not necessarily correctness.

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Cite this review

Pith. "Pith review of Approximately Optimal Search on a Higher-dimensional Sliding Puzzle." pith.science (2026). https://pith.science/paper/I57DUTOD

@misc{pith2026241201937,
  author       = {Pith},
  title        = {Pith review of: Approximately Optimal Search on a Higher-dimensional Sliding Puzzle},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I57DUTOD}},
  note         = {Machine review of arXiv:2412.01937}
}

abstract

Higher-dimensional sliding puzzles are constructed on the vertices of a $d$-dimensional hypercube, where $2^d-l$ vertices are distinctly coloured. Rings with the same colours are initially set randomly on the vertices of the hypercube. The goal of the puzzle is to move each of the $2^d-l$ rings to pre-defined target vertices on the cube. In this setting, the $k$-rule constraint represents a generalisation of edge collision for the movement of colours between vertices, allowing movement only when a hypercube face of dimension $k$ containing a ring is completely free of other rings. Starting from an initial configuration, what is the minimum number of moves needed to make ring colours match the vertex colours? An algorithm that provides us with such a number is called God's algorithm. When such an algorithm exists, it does not have a polynomial time complexity, at least in the case of the 15-puzzle corresponding to $k=1$ in the cubical puzzle. This paper presents a comprehensive computational study of different scenarios of the higher-dimensional puzzle. A benchmark of three computational techniques, an exact algorithm (the A* search) and two approximately optimal search techniques (an evolutionary algorithm (EA) and reinforcement learning (RL)) is presented in this work. The experiments show that all three methods can successfully solve the puzzle of dimension three for different face dimensions and across various difficulty levels. When the dimension increases, the A* search fails, and RL and EA methods can still provide a generally acceptable solution, i.e. a distribution of a number of moves with a median value of less than $30$. Overall, the EA method consistently requires less computational time, while failing in most cases to minimise the number of moves for the puzzle dimensions $d=4$ and $d=5$.

Figures

Figures reproduced from arXiv: 2412.01937 by the authors.

**Figure 1.** An illustration of the classical 15-puzzle and the high-dimensional puzzle of dimension d = 3 for the difficulty level 0. On the top rows is a classic example of the 15-puzzle. Starting from configuration (a), the puzzle consists of numbered square tiles which can be slid into a frame using an empty slot. The object uses the space to slide all tiles where they belong (b), the target configuration. The bottom rows sh… view at source ↗

**Figure 2.** RL Performances: distributions of the number of moves across different difficulty levels for dimensions d = 3, 4 with different face dimensions k = {1, · · · , d − 1}. The number of moves increases with the face dimension and across difficulty levels, with a significant gap between level 4 and the other levels. Compared to RL and EA methods, A* search outperforms success in finding the optimal solution for dimension… view at source ↗

**Figure 3.** EA Performances: distribution of the number of moves for different levels of difficulties and with puzzle parameters k = 2, d = 3 and l = 4. The minimum, maximum and median number of moves increases with difficulty. The minimum number of moves for different difficulty levels is 6, 7, and 10, with a frequency of 14, 2, and 3 out of 150 runs. configurations (one move) but occasionally generate configurations far away … view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: EA vs. RL number of moves distributions across different difficulty levels for the face dimension k = 2. (a) For the puzzles of dimension d = 3, the EA number of move distributions have a wider spread of values, while RL distributions exhibit a more concentrated distri…

**Figure 5.** Figure 5: A* search CPU time plotted against the difficulty levels for different face dimensions. The dot points are different measurements, and the solid lines connect the medians. The A* algorithm consistently requires less computation time for smaller values of k across all l…

**Figure 6.** Figure 6: A* search vs. Comparison of CPU times for A*, RL, and EA methods across various puzzle difficulty levels for all face dimensions. The results indicate that EA outperforms A* and RL methods, especially for higher difficulty levels, with reduced processing time. (a) For …

**Figure 7.** Figure 7: EA: The success rate and the mean number of generations needed to reach the target configuration T with respect to the mutation parameter c (respectively in blue and orange). Performance degrades with the increase of c, and the number of generations increases with c. 2…

**Figure 8.** Figure 8: EA: 6 moves’ frequency vs. mutation parameter c and the frequency of the number of moves for the best parameter c ∗ for the higher-dimension puzzle with parameter d = 3, k = 2 and l = 4. (a) The 6 move frequency with respect to the mutation parameter c. Based on the 6 …

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Reference graph

Works this paper leans on

38 extracted references · 34 canonical work pages · cited by 1 Pith paper

[4]

Higher-dimensional cubical sliding puzzles

Moritz Beyer, Stefano Mereta, ´Erika Rold´ an, and Peter Voran. Higher-dimensional cubical sliding puzzles. arXiv Preprint arXiv:2307.14143 , 2023

work page Pith review arXiv 2023
[1]

Graph puzzles, homotopy, and the alternating group

Richard M Wilson. Graph puzzles, homotopy, and the alternating group. Journal of Combinatorial Theory, Series B , 16(1):86–96, 1974

work page 1974
[2]

Discrete configuration spaces of squares and hexagons

Hannah Alpert. Discrete configuration spaces of squares and hexagons. Journal of Applied and Compu- tational Topology, 4(2):263–280, 2020

work page 2020
[3]

Parity Property of Hexagonal Sliding Puzzles

Ray Karpman and ´Erika Rold´ an. Parity property of hexagonal sliding puzzles. arXiv preprint arXiv:2201.00919, 2022

work page Pith review arXiv 2022
[5]

Finding the shortest move-sequence in the graph-generalized 15-puzzle is NP-hard

Oded Goldreich. Finding the shortest move-sequence in the graph-generalized 15-puzzle is NP-hard. In Studies in complexity and cryptography. Miscellanea on the interplay between randomness and computa- tion, pages 1–5. Springer, 2011

work page 2011
[6]

A formal basis for the heuristic determination of minimum cost paths

Peter E Hart, Nils J Nilsson, and Bertram Raphael. A formal basis for the heuristic determination of minimum cost paths. IEEE Transactions on Systems Science and Cybernetics , 4(2):100–107, 1968

work page 1968
[7]

Reinforcement Learning: An Introduction

Alex M Andrew. Reinforcement Learning: An Introduction. Robotica, 17(2):229–235, 1999. 17

work page 1999
[8]

Markov games as a framework for multi-agent reinforcement learning

Michael L Littman. Markov games as a framework for multi-agent reinforcement learning. In Machine Learning Proceedings, pages 157–163. Elsevier, 1994

work page 1994

Show all 38 references

[9]

Reinforcement learning and approximate dynamic programming for feedback control

Frank L Lewis and Derong Liu. Reinforcement learning and approximate dynamic programming for feedback control. John Wiley & Sons, 2013

2013
[10]

Approximate Dynamic Programming: Solving the curses of dimensionality , volume

Warren B Powell. Approximate Dynamic Programming: Solving the curses of dimensionality , volume
[11]

Reinforcement learning: An introduction

Richard S Sutton. Reinforcement learning: An introduction. A Bradford Book, 2018

2018
[12]

Simulation-based optimization, volume 62

Abhijit Gosavi. Simulation-based optimization, volume 62. Springer, 2015

2015
[13]

A comprehensive survey of multiagent rein- forcement learning

Lucian Busoniu, Robert Babuska, and Bart De Schutter. A comprehensive survey of multiagent rein- forcement learning. IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews), 38(2):156–172, 2008

2008
[14]

Nature-inspired optimization algorithms, volume 2

Xin-She Yang. Nature-inspired optimization algorithms, volume 2. Elsevier, 2020

2020
[15]

Optimal dynamic treatment regimes

Susan A Murphy. Optimal dynamic treatment regimes. Journal of the Royal Statistical Society Series B: Statistical Methodology, 65(2):331–355, 2003

2003
[16]

A Markovian Decision Process

Richard Bellman. A Markovian Decision Process. Journal of Mathematics and Mechanics , pages 679– 684, 1957

1957
[17]

Reinforcement learning for combinatorial optimization: A survey

Nina Mazyavkina, Sergey Sviridov, Sergei Ivanov, and Evgeny Burnaev. Reinforcement learning for combinatorial optimization: A survey. Computers & Operations Research, 134:105400, 2021

2021
[18]

Neural combinatorial optimization with reinforcement learning

Irwan Bello, Hieu Pham, Quoc V Le, Mohammad Norouzi, and Samy Bengio. Neural combinatorial optimization with reinforcement learning. arXiv Preprint arXiv:1611.09940 , 2016

2016 arXiv
[19]

Reinforcement learning for solving the vehicle routing problem

Mohammadreza Nazari, Afshin Oroojlooy, Lawrence Snyder, and Martin Tak´ ac. Reinforcement learning for solving the vehicle routing problem. Advances in Neural Information Processing Systems , 31, 2018

2018
[20]

Exploratory combinatorial opti- mization with reinforcement learning

Thomas Barrett, William Clements, Jakob Foerster, and Alex Lvovsky. Exploratory combinatorial opti- mization with reinforcement learning. In Proceedings of the AAAI Conference on Artificial Intelligence , volume 34, pages 3243–3250, 2020

2020
[21]

Combinatorial optimization with graph convolutional networks and guided tree search

Zhuwen Li, Qifeng Chen, and Vladlen Koltun. Combinatorial optimization with graph convolutional networks and guided tree search. Advances in Neural Information Processing Systems , 31, 2018

2018
[22]

Learning combinatorial optimization algorithms over graphs

Elias Khalil, Hanjun Dai, Yuyu Zhang, Bistra Dilkina, and Le Song. Learning combinatorial optimization algorithms over graphs. Advances in Neural Information Processing Systems , 30, 2017

2017
[23]

Learning local search heuristics for boolean satisfiability

Emre Yolcu and Barnab´ as P´ oczos. Learning local search heuristics for boolean satisfiability. Advances in Neural Information Processing Systems , 32, 2019

2019
[24]

Deep-pack: A vision-based 2d online bin packing algorithm with deep reinforcement learning

Olyvia Kundu, Samrat Dutta, and Swagat Kumar. Deep-pack: A vision-based 2d online bin packing algorithm with deep reinforcement learning. In 2019 28th IEEE International Conference on Robot and Human Interactive Communication (RO-MAN) , pages 1–7. IEEE, 2019

2019
[25]

Attend2pack: Bin packing through deep reinforcement learning with attention

Jingwei Zhang, Bin Zi, and Xiaoyu Ge. Attend2pack: Bin packing through deep reinforcement learning with attention. arXiv preprint arXiv:2107.04333 , 2021

2021 arXiv
[26]

John H. Holland. Adaptation in Natural and Artificial Systems: an introductory analysis with applica- tions to biology, control, and artificial intelligence . The MIT Press London, 1992

1992
[27]

aRNAque: an evolutionary algorithm for inverse pseudoknotted RNA folding inspired by L´ evy flights.BMC Bioinformatics , 23(1):335, 2022

Nono SC Merleau and Matteo Smerlak. aRNAque: an evolutionary algorithm for inverse pseudoknotted RNA folding inspired by L´ evy flights.BMC Bioinformatics , 23(1):335, 2022

2022
[28]

A simple evolutionary algorithm guided by local mutations for an efficient rna design

Nono SC Merleau and Matteo Smerlak. A simple evolutionary algorithm guided by local mutations for an efficient rna design. In Proceedings of the Genetic and Evolutionary Computation conference , pages 1027–1034, 2021. 18

2021
[29]

Evolutionary solution for the RNA design problem

Ali Esmaili-Taheri, Mohammad Ganjtabesh, and Morteza Mohammad-Noori. Evolutionary solution for the RNA design problem. Bioinformatics, 30(9):1250–1258, 2014

2014
[30]

ERD: a fast and reliable tool for RNA design including constraints

Ali Esmaili-Taheri and Mohammad Ganjtabesh. ERD: a fast and reliable tool for RNA design including constraints. BMC Bioinformatics , 16(1):20, 2015

2015
[31]

RnaPredict—an evolutionary algorithm for RNA secondary structure prediction

Kay Wiese, Alain Deschenes, and Andrew Hendriks. RnaPredict—an evolutionary algorithm for RNA secondary structure prediction. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 5(1):25–41, 2008

2008
[32]

An introduction to genetic algorithms

Melanie Mitchell. An introduction to genetic algorithms . MIT Press, 1998

1998
[33]

An evolutionary algorithm for the vehicle routing problem with route balancing

Nicolas Jozefowiez, Fr´ ed´ eric Semet, and El-Ghazali Talbi. An evolutionary algorithm for the vehicle routing problem with route balancing. European Journal of Operational Research, 195(3):761–769, 2009

2009
[34]

A hybrid genetic algorithm for the vehicle routing problem with simultaneous pickup and delivery

Fanggeng Zhao, Dong Mei, Jiangsheng Sun, and Weimin Liu. A hybrid genetic algorithm for the vehicle routing problem with simultaneous pickup and delivery. In 2009 Chinese Control and Decision Conference, pages 3928–3933. IEEE, 2009

2009
[35]

Graph puzzles, homotopy, and the alternating group

Richard M Wilson. Graph puzzles, homotopy, and the alternating group. Journal of Combinatorial Theory, Series B , 16:88–96, 1974

1974
[36]

Artificial Intelligence: A Modern Approach (2nd ed.)

Stuart Russell and Peter Norvig. Artificial Intelligence: A Modern Approach (2nd ed.) . Prentice Hall, 2003

2003
[37]

Reinforcement learning-assisted evolutionary algorithm: A survey and research opportunities

Yanjie Song, Yutong Wu, Yangyang Guo, Ran Yan, Ponnuthurai Nagaratnam Suganthan, Yue Zhang, Witold Pedrycz, Swagatam Das, Rammohan Mallipeddi, Oladayo Solomon Ajani, et al. Reinforcement learning-assisted evolutionary algorithm: A survey and research opportunities. Swarm and E...

2024
[703]

John Wiley & Sons, 2007

2007

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [4]

Higher-dimensional cubical sliding puzzles

Moritz Beyer, Stefano Mereta, ´Erika Rold´ an, and Peter Voran. Higher-dimensional cubical sliding puzzles. arXiv Preprint arXiv:2307.14143 , 2023

work page Pith review arXiv 2023

[2] [1]

Graph puzzles, homotopy, and the alternating group

Richard M Wilson. Graph puzzles, homotopy, and the alternating group. Journal of Combinatorial Theory, Series B , 16(1):86–96, 1974

work page 1974

[3] [2]

Discrete configuration spaces of squares and hexagons

Hannah Alpert. Discrete configuration spaces of squares and hexagons. Journal of Applied and Compu- tational Topology, 4(2):263–280, 2020

work page 2020

[4] [3]

Parity Property of Hexagonal Sliding Puzzles

Ray Karpman and ´Erika Rold´ an. Parity property of hexagonal sliding puzzles. arXiv preprint arXiv:2201.00919, 2022

work page Pith review arXiv 2022

[5] [5]

Finding the shortest move-sequence in the graph-generalized 15-puzzle is NP-hard

Oded Goldreich. Finding the shortest move-sequence in the graph-generalized 15-puzzle is NP-hard. In Studies in complexity and cryptography. Miscellanea on the interplay between randomness and computa- tion, pages 1–5. Springer, 2011

work page 2011

[6] [6]

A formal basis for the heuristic determination of minimum cost paths

Peter E Hart, Nils J Nilsson, and Bertram Raphael. A formal basis for the heuristic determination of minimum cost paths. IEEE Transactions on Systems Science and Cybernetics , 4(2):100–107, 1968

work page 1968

[7] [7]

Reinforcement Learning: An Introduction

Alex M Andrew. Reinforcement Learning: An Introduction. Robotica, 17(2):229–235, 1999. 17

work page 1999

[8] [8]

Markov games as a framework for multi-agent reinforcement learning

Michael L Littman. Markov games as a framework for multi-agent reinforcement learning. In Machine Learning Proceedings, pages 157–163. Elsevier, 1994

work page 1994