 
Summary: Other Complexity Classes and Measures
Eric Allender1
Rutgers University
Michael C. Loui2
University of Illinois at UrbanaChampaign
Kenneth W. Regan3
State University of New York at Buffalo
1 Introduction
In the previous two chapters, we have
· Introduced the basic complexity classes,
· Summarized the known relationships between these classes, and
· Seen how reducibility and completeness can be used to establish tight links between natural
computational problems and complexity classes.
Some natural problems seem not to be complete for any of the complexity classes we have seen
so far. For example, consider the problem of taking as input a graph G and a number k, and
deciding whether k is exactly the length of the shortest traveling salesperson's tour. This is clearly
related to the TSP problem discussed in Chapter 23, but in contrast to TSP, it seems not to belong
to NP, and also seems not to belong to coNP.
To classify and understand this and other problems, we will introduce a few more complexity
classes. We cannot discuss all of the classes that have been studiedthere are further pointers to
