m Solutions Good, m-1 Solutions Better

Authors

Luc Longpre, The University of Texas at El PasoFollow
William Gasarch
G. W. Walster
Vladik Kreinovich, The University of Texas at El PasoFollow

Publication Date

8-2007

Comments

UTEP-CS-00-40b.

Published in Applied Mathematical Sciences, 2008, Vol. 2, No. 5, pp. 223-239.

Abstract

One of the main objectives of theoretical research in computational complexity and feasibility is to explain experimentally observed difference in complexity.

Empirical evidence shows that the more solutions a system of equations has, the more difficult it is to solve it. Similarly, the more global maxima a continuous function has, the more difficult it is to locate them. Until now, these empirical facts have been only partially formalized: namely, it has been shown that problems with two or more solutions are more difficult to solve than problems with exactly one solution. In this paper, we extend this result and show that for every m, problems with exactly m solutions are more difficult to solve than problems with m-1 solutions.

Rephrasing Orwell's "Four legs good, two legs better", we can describe this result as "m solutions good, m-1 solutions better".