Other Complexity Classes and Measures

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 problem discussed in Chapter 28, Section 3, but in contrast to , it seems not to belong to NP, and also seems not to belong to co-NP. 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 studied—there are further pointers to the literature at the end of this chapter. Our goal is to describe some of the most important classes, such as those defined by probabilistic and interactive computation. A common theme is that the new classes arise from the interaction of complexity theory with other fields, such as randomized algorithms, formal logic, combinatorial optimization, and matrix algebra. Complexity theory provides a common formal language for analyzing computational performance in these areas. Other examples can be found in other chapters of this Handbook. 1Supported by the National Science Foundation under Grant CCR-9509603. Portions of this work were performed while a visiting scholar at the Institute of Mathematical Sciences, Madras, India. 2Supported by the National Science Foundation under Grant CCR-9315696. 3Supported by the National Science Foundation under Grant CCR-9409104.