Beating Brute Force for Compression Problems
Authors: 
Shuichi Hirahara, Rahul Ilango, and 
R. Ryan WilliamsAuthors Info & Claims
Shuichi Hirahara, Rahul Ilango, and R. Ryan WilliamsAuthors Info & ClaimsJune 2024
Pages 659 - 670
Abstract
A compression problem is defined with respect to an efficient encoding function f; given a string x, our task is to find the shortest y such that f(y) = x. The obvious brute-force algorithm for solving this compression task on n-bit strings runs in time O(2ℓ · t(n)), where ℓ is the length of the shortest description y and t(n) is the time complexity of f when it prints n-bit output. We prove that every compression problem has a Boolean circuit family which finds short descriptions more efficiently than brute force. In particular, our circuits have size 24 ℓ / 5 · poly(t(n)), which is significantly more efficient for all ℓ ≫ log(t(n)). Our construction builds on Fiat-Naor’s data structure for function inversion [SICOMP 1999]: we show how to carefully modify their data structure so that it can be nontrivially implemented using Boolean circuits, and we show how to utilize hashing so that the circuit size is only exponential in the description length. As a consequence, the Minimum Circuit Size Problem for generic fan-in two circuits of size s(n) on truth tables of size 2n can be solved by circuits of size 24/5 · w + o(w) · poly(2n), where w = s(n) log2(s(n) + n). This improves over the brute-force approach of trying all possible size-s(n) circuits for all s(n) ≥ n. Similarly, the task of computing a short description of a string x when its t-complexity is at most ℓ, has circuits of size 24/5 ℓ · poly(t). We also give nontrivial circuits for computing Kt complexity on average, and for solving NP relations with “compressible” instance-witness pairs.
References
[1]
Eric Allender, Harry Buhrman, Michal Koucký, Dieter van Melkebeek, and Detlef Ronneburger. 2006. Power from Random Strings. SIAM J. Comput., 35, 6 (2006), 1467–1493. https://doi.org/10.1137/050628994
[2]
Noga Alon, Raphael Yuster, and Uri Zwick. 1995. Color-Coding. J. ACM, 42, 4 (1995), 844–856. https://doi.org/10.1145/210332.210337
[3]
Luis Antunes, Lance Fortnow, Dieter van Melkebeek, and N. V. Vinodchandran. 2006. Computational depth: Concept and applications. Theor. Comput. Sci., 354, 3 (2006), 391–404. https://doi.org/10.1016/j.tcs.2005.11.033
[4]
Sanjeev Arora and Boaz Barak. 2009. Computational Complexity - A Modern Approach. Cambridge University Press. isbn:978-0-521-42426-4
[5]
Daniel J. Bernstein and Tanja Lange. 2013. Non-uniform Cracks in the Concrete: The Power of Free Precomputation. In Proceedings of the International Conference on the Theory and Application of Cryptology and Information Security (ASIACRYPT). 321–340. https://doi.org/10.1007/978-3-642-42045-0_17
[6]
Andrej Bogdanov and Luca Trevisan. 2006. Average-Case Complexity. Foundations and Trends in Theoretical Computer Science, 2, 1 (2006), https://doi.org/10.1561/0400000004
[7]
Marco L. Carmosino, Russell Impagliazzo, Valentine Kabanets, and Antonina Kolokolova. 2016. Learning Algorithms from Natural Proofs. In Proceedings of the Conference on Computational Complexity (CCC). 10:1–10:24. https://doi.org/10.4230/LIPIcs.CCC.2016.10
[8]
Dror Chawin, Iftach Haitner, and Noam Mazor. 2020. Lower Bounds on the Time/Memory Tradeoff of Function Inversion. In Proceedings of the Theory of Cryptography Conference (TCC). 305–334. https://doi.org/10.1007/978-3-030-64381-2_11
[9]
Lijie Chen, Shuichi Hirahara, Igor Carboni Oliveira, Ján Pich, Ninad Rajgopal, and Rahul Santhanam. 2022. Beyond Natural Proofs: Hardness Magnification and Locality. J. ACM, 69, 4 (2022), 25:1–25:49. https://doi.org/10.1145/3538391
[10]
Lijie Chen, Ce Jin, and R. Ryan Williams. 2019. Hardness Magnification for all Sparse NP Languages. In Proceedings of the Symposium on Foundations of Computer Science (FOCS). 1240–1255. https://doi.org/10.1109/FOCS.2019.00077
[11]
Lijie Chen and Roei Tell. 2020. Simple and fast derandomization from very hard functions: Eliminating randomness at almost no cost. Electron. Colloquium Comput. Complex., 27 (2020), 148.
[12]
Ruiwen Chen and Valentine Kabanets. 2016. Correlation bounds and #SAT algorithms for small linear-size circuits. Theor. Comput. Sci., 654 (2016), 2–10. https://doi.org/10.1016/J.TCS.2016.05.005
[13]
Mahdi Cheraghchi, Shuichi Hirahara, Dimitrios Myrisiotis, and Yuichi Yoshida. 2021. One-Tape Turing Machine and Branching Program Lower Bounds for MCSP. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS). 23:1–23:19. https://doi.org/10.4230/LIPIcs.STACS.2021.23
[14]
Henry Corrigan-Gibbs and Dmitry Kogan. 2019. The Function-Inversion Problem: Barriers and Opportunities. In Proceedings of the Theory of Cryptography Conference (TCC). 393–421. https://doi.org/10.1007/978-3-030-36030-6_16
[15]
Anindya De, Luca Trevisan, and Madhur Tulsiani. 2010. Time Space Tradeoffs for Attacks against One-Way Functions and PRGs. In Advances in Cryptology - CRYPTO 2010, 30th Annual Cryptology Conference, Santa Barbara, CA, USA, August 15-19, 2010. Proceedings. 649–665. https://doi.org/10.1007/978-3-642-14623-7_35
[16]
Pavel Dvorák, Michal Koucký, Karel Král, and Veronika Slívová. 2021. Data Structures Lower Bounds and Popular Conjectures. In Proceedings of the European Symposium on Algorithms (ESA). 39:1–39:15. https://doi.org/10.4230/LIPICS.ESA.2021.39
[17]
Amos Fiat and Moni Naor. 1999. Rigorous Time/Space Trade-offs for Inverting Functions. SIAM J. Comput., 29, 3 (1999), 790–803. https://doi.org/10.1137/S0097539795280512
[18]
Charles M. Fiduccia. 1972. Polynomial Evaluation via the Division Algorithm: The Fast Fourier Transform Revisited. In Proceedings of the Symposium on Theory of Computing (STOC). 88–93. https://doi.org/10.1145/800152.804900
[19]
Gudmund Skovbjerg Frandsen and Peter Bro Miltersen. 2005. Reviewing bounds on the circuit size of the hardest functions. Inf. Process. Lett., 95, 2 (2005), 354–357. https://doi.org/10.1016/j.ipl.2005.03.009
[20]
Alexander Golovnev, Siyao Guo, Thibaut Horel, Sunoo Park, and Vinod Vaikuntanathan. 2020. Data structures meet cryptography: 3SUM with preprocessing. In Proceedings of the Symposium on Theory of Computing (STOC). 294–307. https://doi.org/10.1145/3357713.3384342
[21]
Alexander Golovnev, Siyao Guo, Spencer Peters, and Noah Stephens-Davidowitz. 2023. Revisiting Time-Space Tradeoffs for Function Inversion. In Proceedings of the International Cryptology Conference (CRYPTO). 453–481. https://doi.org/10.1007/978-3-031-38545-2_15
[22]
Alexander Golovnev, Alexander S. Kulikov, Alexander V. Smal, and Suguru Tamaki. 2016. Circuit Size Lower Bounds and #SAT Upper Bounds Through a General Framework. In Proceedings of the International Symposium on Mathematical Foundations of Computer Science (MFCS). 45:1–45:16. https://doi.org/10.4230/LIPICS.MFCS.2016.45
[23]
Martin E. Hellman. 1980. A cryptanalytic time-memory trade-off. IEEE Trans. Inf. Theory, 26, 4 (1980), 401–406. https://doi.org/10.1109/TIT.1980.1056220
[24]
Shuichi Hirahara. 2018. Non-Black-Box Worst-Case to Average-Case Reductions within NP. In Proceedings of the Symposium on Foundations of Computer Science (FOCS). 247–258. https://doi.org/10.1109/FOCS.2018.00032
[25]
Shuichi Hirahara. 2020. Non-Disjoint Promise Problems from Meta-Computational View of Pseudorandom Generator Constructions. In Proceedings of the Computational Complexity Conference (CCC). 20:1–20:47. https://doi.org/10.4230/LIPIcs.CCC.2020.20
[26]
Shuichi Hirahara. 2021. Average-case hardness of NP from exponential worst-case hardness assumptions. In Proceedings of the Symposium on Theory of Computing (STOC). 292–302. https://doi.org/10.1145/3406325.3451065
[27]
Shuichi Hirahara. 2022. NP-Hardness of Learning Programs and Partial MCSP. In Proceedings of the Symposium on Foundations of Computer Science (FOCS). 968–979. https://doi.org/10.1109/FOCS54457.2022.00095
[28]
Shuichi Hirahara and Mikito Nanashima. 2021. On Worst-Case Learning in Relativized Heuristica. In Proceedings of the Symposium on Foundations of Computer Science (FOCS). 751–758. https://doi.org/10.1109/FOCS52979.2021.00078
[29]
Christopher M. Homan and Mayur Thakur. 2003. One-way permutations and self-witnessing languages. J. Comput. Syst. Sci., 67, 3 (2003), 608–622. https://doi.org/10.1016/S0022-0000(03)00068-0
[30]
Rahul Ilango. 2023. SAT Reduces to the Minimum Circuit Size Problem with a Random Oracle. In Proceedings of the Symposium on Foundations of Computer Science (FOCS). 733–742. https://doi.org/10.1109/FOCS57990.2023.00048
[31]
Rahul Ilango, Hanlin Ren, and Rahul Santhanam. 2022. Robustness of average-case meta-complexity via pseudorandomness. In Proceedings of the Symposium on Theory of Computing (STOC). 1575–1583. https://doi.org/10.1145/3519935.3520051
[32]
Russell Impagliazzo and Leonid A. Levin. 1990. No Better Ways to Generate Hard NP Instances than Picking Uniformly at Random. In Proceedings of the Symposium on Foundations of Computer Science (FOCS). 812–821. https://doi.org/10.1109/FSCS.1990.89604
[33]
Russell Impagliazzo and Avi Wigderson. 1997. P = BPP if E Requires Exponential Circuits: Derandomizing the XOR Lemma. In Proceedings of the Symposium on the Theory of Computing (STOC). 220–229. https://doi.org/10.1145/258533.258590
[34]
Leonid A. Levin. 1986. Average Case Complete Problems. SIAM J. Comput., 15, 1 (1986), 285–286. https://doi.org/10.1137/0215020
[35]
Ming Li and Paul M. B. Vitányi. 2019. An Introduction to Kolmogorov Complexity and Its Applications, 4th Edition. Springer. isbn:978-3-030-11297-4 https://doi.org/10.1007/978-3-030-11298-1
[36]
Yanyi Liu and Rafael Pass. 2020. On One-way Functions and Kolmogorov Complexity. In Proceedings of the Symposium on Foundations of Computer Science (FOCS). 1243–1254.
[37]
Yanyi Liu and Rafael Pass. 2021. On the Possibility of Basing Cryptography on EXP¬ = BPP. In Proceedings of the International Cryptology Conference (CRYPTO). 11–40. https://doi.org/10.1007/978-3-030-84242-0_2
[38]
Noam Mazor and Rafael Pass. 2024. The Non-Uniform Perebor Conjecture for Time-Bounded Kolmogorov Complexity Is False. In Proceedings of the Innovations in Theoretical Computer Science Conference (ITCS). 80:1–80:20. https://doi.org/10.4230/LIPICS.ITCS.2024.80
[39]
Dylan M. McKay, Cody D. Murray, and R. Ryan Williams. 2019. Weak lower bounds on resource-bounded compression imply strong separations of complexity classes. In Proceedings of the Symposium on Theory of Computing (STOC). 1215–1225. https://doi.org/10.1145/3313276.3316396
[40]
Igor Carboni Oliveira, Ján Pich, and Rahul Santhanam. 2019. Hardness Magnification near State-Of-The-Art Lower Bounds. In Proceedings of the Computational Complexity Conference (CCC). 27:1–27:29. https://doi.org/10.4230/LIPIcs.CCC.2019.27
[41]
Igor Carboni Oliveira and Rahul Santhanam. 2018. Hardness Magnification for Natural Problems. In Proceedings of the Symposium on Foundations of Computer Science (FOCS). 65–76.
[42]
Wolfgang J. Paul. 1976. Realizing Boolean Functions on Disjoint sets of Variables. Theor. Comput. Sci., 2, 3 (1976), 383–396. https://doi.org/10.1016/0304-3975(76)90089-X
[43]
Nicholas Pippenger and Michael J. Fischer. 1979. Relations Among Complexity Measures. J. ACM, 26, 2 (1979), 361–381. https://doi.org/10.1145/322123.322138
[44]
Alexander A. Razborov and Steven Rudich. 1997. Natural Proofs. J. Comput. Syst. Sci., 55, 1 (1997), 24–35. https://doi.org/10.1006/jcss.1997.1494
[45]
Hanlin Ren and Rahul Santhanam. 2021. Hardness of KT Characterizes Parallel Cryptography. In Proceedings of the Computational Complexity Conference (CCC). 35:1–35:58. https://doi.org/10.4230/LIPIcs.CCC.2021.35
[46]
Boris A. Trakhtenbrot. 1984. A Survey of Russian Approaches to Perebor (Brute-Force Searches) Algorithms. IEEE Annals of the History of Computing, 6, 4 (1984), 384–400. https://doi.org/10.1109/MAHC.1984.10036
[47]
Luca Trevisan. 2010. The Program-Enumeration Bottleneck in Average-Case Complexity Theory. In Proceedings of the Conference on Computational Complexity (CCC). 88–95. https://doi.org/10.1109/CCC.2010.18
[48]
Leslie G. Valiant and Vijay V. Vazirani. 1986. NP is as Easy as Detecting Unique Solutions. Theor. Comput. Sci., 47, 3 (1986), 85–93. https://doi.org/10.1016/0304-3975(86)90135-0
[49]
Joachim von zur Gathen and Jürgen Gerhard. 2013. Modern Computer Algebra (3. ed.). Cambridge University Press. isbn:978-1-107-03903-2
[50]
Ryan Williams. 2013. Improving Exhaustive Search Implies Superpolynomial Lower Bounds. SIAM J. Comput., 42, 3 (2013), 1218–1244. https://doi.org/10.1137/10080703X
[51]
Andrew Chi-Chih Yao. 1990. Coherent Functions and Program Checkers (Extended Abstract). In Proceedings of the Symposium on Theory of Computing (STOC). 84–94. https://doi.org/10.1145/100216.100226
Index Terms
- Beating Brute Force for Compression Problems
Recommendations
Beating brute force for (quantified) satisfiability of circuits of bounded treewidth
SODA '18: Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete AlgorithmsWe investigate the algorithmic properties of circuits of bounded treewidth. Here the treewidth of a circuit C is defined as the treewidth of the underlying undirected graph of C, after the vertices corresponding to input gates have been removed. Thus, ...
Beating Treewidth for Average-Case Subgraph Isomorphism
AbstractFor any fixed graph G, the subgraph isomorphism problem asks whether an n-vertex input graph has a subgraph isomorphic to G. A well-known algorithm of Alon et al. (J ACM 42(4):844–856, 1995. https://doi.org/10.1145/210332.210337) efficiently ...
Super-linear gate and super-quadratic wire lower bounds for depth-two and depth-three threshold circuits
STOC '16: Proceedings of the forty-eighth annual ACM symposium on Theory of ComputingIn order to formally understand the power of neural computing, we first need to crack the frontier of threshold circuits with two and three layers, a regime that has been surprisingly intractable to analyze. We prove the first super-linear gate lower ...
Comments
Information & Contributors
Information
Published In
June 2024
2049 pages
ISBN:9798400703836
DOI:10.1145/3618260
- General Chairs:
- Bojan Mohar,
- Igor Shinkar,
- Program Chair:
- Ryan O'Donnell
Copyright © 2024 Owner/Author.
This work is licensed under a Creative Commons Attribution International 4.0 License.
Sponsors
Publisher
Association for Computing Machinery
New York, NY, United States
Publication History
Published: 11 June 2024
Check for updates
Author Tags
Qualifiers
- Research-article
Funding Sources
- JST, PRESTO
- NSF Graduate Fellowship
- NSF
Conference
STOC '24
Sponsor:
Acceptance Rates
Overall Acceptance Rate 1,469 of 4,586 submissions, 32%
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 75Total Downloads
- Downloads (Last 12 months)75
- Downloads (Last 6 weeks)75
Other Metrics
Citations
View Options
Get Access
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign in