Instance complexity

We introduce a measure for the computational complexity of mdiwdual instances of a decision problem and study some of Its properties. The instance complexity of a string ~ with respect to a set A and time bound t, ict(x : A). is defined as the size of the smallest special-case program for A that run> m time t,decides x correctly, and makes no mistakes on other strings (“don’t know” answers are permitted). We prove that a set A is m P if and only if there exist a polynomial t and a constant c such that ic’(x : A) < c for all X; on the other hand, If A ]s NP-hard and P # NP, then for all polynomials t and constants c. lc’(~ : A) > c log I ~ I for ]nfimtely many x. Obserwng that Kf(x), the t-bounded Kolmogorov complexity of x, N roughly an upper bound on ]Ct(.t : A), we proceed to investigate the existence of mdiwdually hard problem Instances. ].e , strings whose instance complexity E close to their Kolmogorov complexity. We prove that if t(n)z n is a time-constructible function and A 1s a recurswe set not in DTIME(t), there then exist a constant c and mfimtely many I such that ic’(x : ,4) z K’ (x) – c. for some Prehmmary versions of parts of this work have appeared under the titles “What 1sa hard instance of a computational problem?” m Proceedings of tize Conference on Structare m Cornplexm Theory (Berkeley, Calif., June i 986), and “On the instance complexity of NP-hard problems” in Procecduzgs of the 5tk .4nrrual Conference on StntctLwe m Cowrpkwty Theory (Barcelona, Spain, July 1990). These Proceedings have been published by Springer-Verlag, Berlin, and IEEE, New York, respectively. The research of P. Orponen was supported by the Academy of Finland, and the research of K. Ko in part by National Science Foundation (NSF) grant CCR 8S-01575. Authors’ current addresses: P. Orponen, Department of Computer Science, Unnerslty of Helsinkl, FIN-0001 4 Helsinki, Finland; K. Ko, Department of Computer Science, State Unwersity of New York at Stony Brook, Stony Brook, NY 11794; U. Schomng, Abteiltrng Theoretische Informatik, Umversltat Ulm, D-89069 Ulm, Germany; O. Watanabe, Department of Computer Science, Tohyo Institute of Technology, Tokyo 152, Japan. Permission to copy without fee all or part of this material IS granted provided that the copies are not made or distributed for duect commercial advantage, the ACM copyright notice and the title of the pubhcdtion and Its date appear, and notice K given that copying 1s by permission of the Association for Computing Machinery. To copy otherwse, or to repubhsh, requmes a fee and/or specific permission. 01994 ACM 0004-5411/94/’0100-0096 $03.50 Journal of the AwocI.tIon for Compuh.g Md.hlncry, Vii 41 No 1, January 1YY4 pp Y6-121 Instance Complexity 97 time bound t‘(n)dependent on the complexity of recognizing A. Under the stronger assumptions that the set A is NP-hard and DEXT # NEXT, we prove that for any polynomia~ t there exist a polynomial f‘ and a constant c such that for infinitely many x, ict(x : A) z K“(x) – c. If A is DEXT-hard, then the same result holds unconditionally. We also prove that there is a set A E DEXT such that for some constant c and all x, ic’xp(x : A) s K’xp (x) – 2 log ZCexPr(x)– C, where exp(n) = 2“ and exp’(n) = cn2zn + c.

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