Polynomial-time self-reducibility: theoretical motivations and practical results^∗

Abstract

Although polynomial-time complexity theory has been formulated in terms of decision problems, polynomial-time decision algorithms generally operate by attempting to construct a solution to an optimization version of the problem at hand. Thus it is that self-reducibility, the process by which a decision algorithm may be used to devise a constructive algorithm, has until now been widely considered a topic of only theoretical interest. Recent fundamental advances in graph theory, however, have made available powerful new nonconstructive tools that can be applied to guarantee membership in P. These tools are nonconstructive at two distinct levels: they neither produce the decision algorithm, establishing only the finiteness of an obstruction set, nor do they reveal whether such a decision algorithm can be of any aid in the construction of a solution. We briefly review and illustrate the use of these tools, and discuss the seemingly formidable task of finding the promised polynomial-time decision algorithms when these new tools apply. Our main focus is on the design of efficient self-reduction strategies, with combinatorial problems drawn from a variety of areas.

Keywords:

• Polynomial-time complexity
• search problems
• self-reducibility
• well-partial-order theory

Notes