All Maximally Complex Problems Allow Simplifying Divide-and-Conquer Approach: Intuitive Explanation of a Somewhat Counterintuitive Ladner's Result

Authors

Olga Kosheleva, The University of Texas at El PasoFollow
Vladik Kreinovich, The University of Texas at El PasoFollow

Publication Date

10-2018

Comments

Technical Report: UTEP-CS-18-71

Abstract

Ladner's 1975 result says that any NP-complete problem -- i.e., in effect, any maximally complex problem -- can be reduced to solving two easier problems. This result sounds counter-intuitive: if a problem is maximally complex, how can it be reduced to simpler ones? In this paper, we provide an intuitive explanation for this result. Our main argument is that since complexity and easiness-to-divide are not perfectly correlated, it is natural to expect that maximally complex problem is not maximally difficult to divide. Our related argument is that -- as this result shows -- NP-completeness is a sufficient but not a necessary condition for a problem to be maximally complex; how to come up with a more adequate notion of complexity is still an open problem.