 
Summary: Reducibility and Completeness
Eric Allender1
Rutgers University
Michael C. Loui2
University of Illinois at UrbanaChampaign
Kenneth W. Regan3
State University of New York at Buffalo
1 Introduction
There is little doubt that the notion of reducibility is the most useful tool that complexity theory
has delivered to the rest of the computer science community.
For most computational problems that arise in realworld applications, such as the Traveling
Salesperson Problem, we still know little about their deterministic time or space complexity. We
cannot now tell whether classes such as P and NP are distinct. And yet, even without such hard
knowledge, it has been useful in practice to take some new problem A whose complexity needs to
be analyzed, and announce that A has roughly the same complexity as the Traveling Salesperson
Problem, by exhibiting efficient ways of reducing each problem to the other. Thus we can say a
lot about problems being equivalent in complexity to each other, even if we cannot pinpoint what
that complexity is.
One reason for this success is that when one partitions the many thousands of realworld com
putational problems into equivalence classes according to the reducibility relation, there are surpris
