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Yet Another Simple Proof of the PCRP Theorem

Author Naoto Ohsaka

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LIPIcs.ICALP.2025.122.pdf
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ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Error-correcting codes
Keywords
  • reconfiguration problems
  • hardness of approximation
  • probabilistic proof systems

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Abstract

The Probabilistically Checkable Reconfiguration Proof (PCRP) theorem, proven by Hirahara and Ohsaka (STOC 2024) [Hirahara and Ohsaka, 2024] and Karthik C. S. and Manurangsi [{Karthik {C. S.}} and Manurangsi, 2023], provides a new PCP-type characterization of PSPACE: A language L is in PSPACE if and only if there exists a probabilistic verifier 𝒱 and a pair of polynomial-time computable proofs π^ini, π^end such that the following hold for every input x:  
- If x ∈ L, then π^ini(x) can be transformed into π^end(x) by repeatedly flipping a single bit of the proof at a time, while making 𝒱(x) to accept every intermediate proof with probability 1. 
- If x ∉ L, then any such transformation induces a proof that is rejected by 𝒱(x) with probability more than 1/2.  The PCRP theorem finds many applications in PSPACE-hardness of approximation for reconfiguration problems.
In this paper, we present an alternative proof of the PCRP theorem that is "simpler" than those of Hirahara and Ohsaka [Hirahara and Ohsaka, 2024] and Karthik C. S. and Manurangsi [Karthik C. S. and Manurangsi, 2023]. Our PCRP system is obtained by combining simple robustization and composition steps in a modular fashion, which renders its analysis more intuitive. The crux of implementing the robustization step is an error-correcting code that enjoys both list decodability and reconfigurability, the latter of which enables to reconfigure between a pair of codewords, while avoiding getting too close to any other codewords.

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Naoto Ohsaka. Yet Another Simple Proof of the PCRP Theorem. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 122:1-122:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.122

Author Details

Naoto Ohsaka
  • CyberAgent, Inc., Tokyo, Japan

References

  1. Sanjeev Arora and Boaz Barak. Computational Complexity: A Modern Approach. Cambridge University Press, 2009. URL: https://doi.org/10.1017/CBO9780511804090.
  2. Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. Journal of the ACM, 45(3):501-555, 1998. URL: https://doi.org/10.1145/278298.278306.
  3. Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs: A new characterization of NP. Journal of the ACM, 45(1):70-122, 1998. URL: https://doi.org/10.1145/273865.273901.
  4. Eli Ben-Sasson, Oded Goldreich, Prahladh Harsha, Madhu Sudan, and Salil Vadhan. Robust PCPs of proximity, shorter PCPs, and applications to coding. SIAM Journal on Computing, 36(4):889-974, 2006. URL: https://doi.org/10.1137/S0097539705446810.
  5. Hans L. Bodlaender, Carla Groenland, Jesper Nederlof, and Céline Swennenhuis. Parameterized problems complete for nondeterministic FPT time and logarithmic space. Information and Computation, 300:105195, 2024. URL: https://doi.org/10.1016/j.ic.2024.105195.
  6. Hans L. Bodlaender, Carla Groenland, and Céline M. F. Swennenhuis. Parameterized complexities of dominating and independent set ceconfiguration. In Proceedings of the International Symposium on Parameterized and Exact Computation (IPEC), pages 9:1-9:16, 2021. URL: https://doi.org/10.4230/LIPIcs.IPEC.2021.9.
  7. Paul Bonsma. The complexity of rerouting shortest paths. Theoretical Computer Science, 510:1-12, 2013. URL: https://doi.org/10.1016/j.tcs.2013.09.012.
  8. Paul Bonsma and Luis Cereceda. Finding paths between graph colourings: PSPACE-completeness and superpolynomial distances. Theoretical Computer Science, 410(50):5215-5226, 2009. URL: https://doi.org/10.1016/j.tcs.2009.08.023.
  9. Nicolas Bousquet, Amer E. Mouawad, Naomi Nishimura, and Sebastian Siebertz. A survey on the parameterized complexity of reconfiguration problems. Computer Science Review, 53:100663, 2024. URL: https://doi.org/10.1016/j.cosrev.2024.100663.
  10. Luis Cereceda, Jan van den Heuvel, and Matthew Johnson. Finding paths between 3-colorings. Journal of Graph Theory, 67(1):69-82, 2011. URL: https://doi.org/10.1002/jgt.20514.
  11. Irit Dinur. The PCP theorem by gap amplification. Journal of the ACM, 54(3):12, 2007. URL: https://doi.org/10.1145/1236457.1236459.
  12. Irit Dinur and Omer Reingold. Assignment testers: Towards a combinatorial proof of the PCP theorem. SIAM Journal on Computing, 36(4):975-1024, 2006. URL: https://doi.org/10.1137/S0097539705446962.
  13. G. David Forney, Jr. Concatenated Codes. PhD thesis, Massachusetts Institute of Technology, 1965. URL: http://hdl.handle.net/1721.1/13449.
  14. Hana Galperin and Avi Wigderson. Succinct representations of graphs. Information and Control, 56(3):183-198, 1983. URL: https://doi.org/10.1016/S0019-9958(83)80004-7.
  15. Parikshit Gopalan, Phokion G. Kolaitis, Elitza Maneva, and Christos H. Papadimitriou. The connectivity of Boolean satisfiability: Computational and structural dichotomies. SIAM Journal on Computing, 38(6):2330-2355, 2009. URL: https://doi.org/10.1137/07070440X.
  16. Venkatesan Guruswami. List Decoding of Error-Correcting Codes. PhD thesis, Massachusetts Institute of Technology, September 2001. URL: http://hdl.handle.net/1721.1/8700.
  17. Venkatesan Guruswami, Atri Rudra, and Madhu Sudan. Essential coding theory, 2019. Draft available at URL: https://cse.buffalo.edu/faculty/atri/courses/coding-theory/book/.
  18. Arash Haddadan, Takehiro Ito, Amer E. Mouawad, Naomi Nishimura, Hirotaka Ono, Akira Suzuki, and Youcef Tebbal. The complexity of dominating set reconfiguration. Theoretical Computer Science, 651:37-49, 2016. URL: https://doi.org/10.1016/j.tcs.2016.08.016.
  19. Robert A. Hearn and Erik D. Demaine. PSPACE-completeness of sliding-block puzzles and other problems through the nondeterministic constraint logic model of computation. Theoretical Computer Science, 343(1-2):72-96, 2005. URL: https://doi.org/10.1016/j.tcs.2005.05.008.
  20. Robert A. Hearn and Erik D. Demaine. Games, Puzzles, and Computation. A K Peters, Ltd., 2009. Google Scholar
  21. Shuichi Hirahara and Naoto Ohsaka. Optimal PSPACE-hardness of approximating set cover reconfiguration. In Proceedings of the International Colloquium on Automata, Languages, and Programming (ICALP), pages 85:1-85:18, 2024. URL: https://doi.org/10.4230/LIPIcs.ICALP.2024.85.
  22. Shuichi Hirahara and Naoto Ohsaka. Probabilistically checkable reconfiguration proofs and inapproximability of reconfiguration problems. In Proceedings of the ACM Symposium on Theory of Computing (STOC), pages 1435-1445, 2024. URL: https://doi.org/10.1145/3618260.3649667.
  23. Shuichi Hirahara and Naoto Ohsaka. Asymptotically optimal inapproximability of maxmin k-cut reconfiguration. In Proceedings of the International Colloquium on Automata, Languages, and Programming (ICALP), 2025. to appear. Google Scholar
  24. Duc A. Hoang. Combinatorial Reconfiguration, 2024. URL: https://reconf.wikidot.com/.
  25. Johan Håstad. Clique is hard to approximate within n^1-ε. Acta Mathematica, 182:105-142, 1999. URL: https://doi.org/10.1007/BF02392825.
  26. Johan Håstad. Some optimal inapproximability results. Journal of the ACM, 48(4):798-859, 2001. URL: https://doi.org/10.1145/502090.502098.
  27. Takehiro Ito, Erik D. Demaine, Nicholas J. A. Harvey, Christos H. Papadimitriou, Martha Sideri, Ryuhei Uehara, and Yushi Uno. On the complexity of reconfiguration problems. Theoretical Computer Science, 412(12-14):1054-1065, 2011. URL: https://doi.org/10.1016/j.tcs.2010.12.005.
  28. Marcin Kamiński, Paul Medvedev, and Martin Milanič. Complexity of independent set reconfigurability problems. Theoretical Computer Science, 439:9-15, 2012. URL: https://doi.org/10.1016/j.tcs.2012.03.004.
  29. Karthik C. S. and Pasin Manurangsi. On inapproximability of reconfiguration problems: PSPACE-hardness and some tight NP-hardness results. Electronic Colloquium on Computational Complexity, TR24-007, 2023. URL: http://eccc.weizmann.ac.il/report/2024/007.
  30. Daniel Lokshtanov and Amer E. Mouawad. The complexity of independent set reconfiguration on bipartite graphs. ACM Transactions on Algorithms, 15(1):7:1-7:19, 2019. URL: https://doi.org/10.1145/3280825.
  31. Amer E. Mouawad, Naomi Nishimura, Venkatesh Raman, Narges Simjour, and Akira Suzuki. On the parameterized complexity of reconfiguration problems. Algorithmica, 78(1):274-297, 2017. URL: https://doi.org/10.1007/s00453-016-0159-2.
  32. Christina M. Mynhardt and Shahla Nasserasr. Reconfiguration of colourings and dominating sets in graphs. In 50 years of Combinatorics, Graph Theory, and Computing, chapter 10, pages 171-191. Chapman and Hall/CRC, 2019. Google Scholar
  33. Naomi Nishimura. Introduction to reconfiguration. Algorithms, 11(4):52, 2018. URL: https://doi.org/10.3390/a11040052.
  34. Naoto Ohsaka. Gap preserving reductions between reconfiguration problems. In Proceedings of the International Symposium on Theoretical Aspects of Computer Science (STACS), pages 49:1-49:18, 2023. URL: https://doi.org/10.4230/LIPIcs.STACS.2023.49.
  35. Naoto Ohsaka. Alphabet reduction for reconfiguration problems. In Proceedings of the International Colloquium on Automata, Languages, and Programming (ICALP), pages 113:1-113:17, 2024. URL: https://doi.org/10.4230/LIPIcs.ICALP.2024.113.
  36. Naoto Ohsaka. Gap amplification for reconfiguration problems. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1345-1366, 2024. URL: https://doi.org/10.1137/1.9781611977912.54.
  37. Naoto Ohsaka. Tight inapproximability of target set reconfiguration. Computing Research Repository, abs/2402.15076, 2024. URL: https://doi.org/10.48550/arXiv.2402.15076.
  38. Naoto Ohsaka. On approximate reconfigurability of label cover. Information Processing Letters, 189:106556, 2025. URL: https://doi.org/10.1016/j.ipl.2024.106556.
  39. Naoto Ohsaka and Tatsuya Matsuoka. Reconfiguration problems on submodular functions. In Proceedings of the ACM International Conference on Web Search and Data Mining (WSDM), pages 764-774, 2022. URL: https://doi.org/10.1145/3488560.3498382.
  40. Christos H. Papadimitriou and Mihalis Yannakakis. A note on succinct representations of graphs. Information and Control, 71(3):181-185, 1986. URL: https://doi.org/10.1016/S0019-9958(86)80009-2.
  41. Irving S. Reed and Gustave Solomon. Polynomial codes over certain finite fields. Journal of the Society for Industrial and Applied Mathematics, 8(2):300-304, 1960. URL: https://doi.org/10.1137/0108018.
  42. Jan van den Heuvel. The complexity of change. In Surveys in Combinatorics 2013, volume 409, pages 127-160. Cambridge University Press, 2013. URL: https://doi.org/10.1017/CBO9781139506748.005.
  43. Marcin Wrochna. Reconfiguration in bounded bandwidth and tree-depth. Journal of Computer and System Sciences, 93:1-10, 2018. URL: https://doi.org/10.1016/j.jcss.2017.11.003.
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