Abstract
The class TFNP, defined by Megiddo and Papadimitriou, consists of multivalued functions with values that are polynomially verifiable and guaranteed to exist. Do we have evidence that such functions are hard, for example, if TFNP is computable in polynomial-time does this imply the polynomial-time hierarchy collapses? By computing a multivalued function in deterministic polynomial-time we mean on every input producing one of the possible values of the function on that input.
We give a relativized negative answer to this question by exhibiting an oracle under which TFNP functions are easy to compute but the polynomial-time hierarchy is infinite. We also show that relative to this same oracle, P≠UP and TFNPNP functions are not computable in polynomial-time with an NP oracle.
References
Baker, T., Gill, J., Solovay, R.: Relativization of P=NP question. SIAM J. Comput. 4(4), 431–442 (1975)
Bellare, M., Goldwasser, S.: The complexity of decision versus search. SIAM J. Comput. 23(1), 97–119 (1994)
Fenner, S., Fortnow, L., Naik, A., Rogers, J.: Inverting onto functions. Inf. Comput. 186, 90–103 (2003)
Fortnow, L., Rogers, J.: Separability and one-way functions. Comput. Complex. 11, 137–157 (2003)
Grollman, J., Selman, A.: Complexity measures for public-key cryptosystems. SIAM J. Comput. 17(2), 309–335 (1988)
Håstad, J.: Almost optimal lower bounds for small depth circuits. Adv. Comput. Res. 5, 143–170 (1989)
Impagliazzo, R., Naor, M.: Decision trees and downward closures. In: Proceedings of the 3rd IEEE Structure in Complexity Theory Conference, pp. 29–38. IEEE, New York (1988)
Kolmogorov, A.N.: Three approaches to the quantitative definition of information. Probl. Inf. Trans. 1(1), 1–7 (1965)
Li, M., Vitányi, P.: An Introduction to Kolmogorov Complexity and Its Applications, 2nd edn. Graduate Texts in Computer Science. Springer, New York (1997)
Megiddo, N., Papadimitriou, C.: On total functions, existence theorems and computational complexity. Theor. Comput. Sci. 81(2), 317–324 (1991)
Meyer, A., Stockmeyer, L.: The equivalence problem for regular expressions with squaring requires exponential space. In: Proceedings of the 13th IEEE Symposium on Switching and Automata Theory, pp. 125–129. IEEE, New York (1972)
Muchnik, A., Vereshchagin, N.: A general method to construct oracles realizing given relationships between complexity classes. Theor. Comput. Sci. 157, 227–258 (1996)
Papadimitriou, C.: Computational Complexity. Addison-Wesley, New York (1994)
Sipser, M.: Borel sets and circuit complexity. In: Proceedings of the 15th ACM Symposium on the Theory of Computing, pp. 61–69. ACM, New York (1983)
Sipser, M.: Introduction to the Theory of Computation. PWS, Boston (1997)
Solomonoff, R.J.: A formal theory of inductive inference, parts 1 and 2. Inf. Control 7, 1–22, 224–254 (1964)
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Buhrman, H., Fortnow, L., Koucký, M. et al. Does the Polynomial Hierarchy Collapse if Onto Functions are Invertible?. Theory Comput Syst 46, 143–156 (2010). https://doi.org/10.1007/s00224-008-9160-8
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DOI: https://doi.org/10.1007/s00224-008-9160-8
Keywords
- Computational complexity
- Polynomial-time hierarchy
- Multi-valued functions
- Kolmogorov complexity