The Instance Complexity Conjecture

The instance complexity of a string x with respect to a set A and time bound t, ict (x : A), is the length of the shortest program for A that runs in time t, decides x correctly, and makes no mistakes on other strings (where \don't know" answers are permitted). The Instance Complexity Conjecture of Ko, Orponen, Schoning, and Watanabe states that for every recursive set A not in P and every polynomial t there is a polynomial t0 and a constant c such that for in nitely many x, ict (x : A) Ct0 (x)?c, where Ct0 (x) is the t0-time bounded Kolmogorov complexity of x. In this paper the conjecture is proved for all recursive tally sets and for all recursive sets which are NP-hard under honest reductions, in particular it holds for all natural NP-hard problems. The method of proof also yields the polynomial-space bounded and the exponential-time bounded versions of the conjecture in full generality. On the other hand, the conjecture itself turns out to be oracle dependent: In any rel...

• Type: Conference
• Year: 1995
• Where: COCO
• Authors: Martin Kummer
• Keywords: Algorithms, COCO 1995, Conjecture, Instance Complexity, Instance Complexity Conjecture