Algorithm Design

552
Chapter 9 PSPACE: A Class of Problems beyond NP
on this problem, covering the state of the art up to a year before IBM’s Deep
Blue finally achieved the goal of defeating the human world champion in a
match.
Planning is a fundamental problem in artificial intelligence; it features
prominently in the text by Russell and Norvig (2002) and is the subject of a
book by Ghallab, Nau, and Traverso (2004). The argument that Planning can
be solved in polynomial space is due to Sa.vitch (1970), who was concerned
with issues in complexity theory rather than the Planning Problem per se.
Notes on the Exercises Exercise 1 is based on a problem we learned from
Maverick Woo and Ryan Williams; Exercise 2 is based on a result of Thomas
Schaefer.
Ch ter
Extending the Limits
Trac~abigity
Although we started the book by studying a number of techniques for solving
problems efficiently, we’ve been looking for a while at classes of problems--
NP-complete and PSPACE-complete problems--for which no efficient solution
is believed to exist. And because of the insights we’ve gained this way,
we’ve implicitly arrived at a two-pronged approach to dealing with new
computational problems we encounter: We try for a while to develop an
efficient algorithm; and if this fails, we then try to prove it NP-complete (or
even PSPACE-complete). Assuming one of the two approaches works out, you
end up either with a solution to the problem (an algorithm), or a potent
"reason" for its difficulty: It is as hard as many of the famous problems in
computer science.
Unfortunately, this strategy will only get you so far. If there is a problem
that people really want your help in solving, they won’t be particularly satisfied
with the resolution that it’s NP-hard I and that they should give up on it. They’ll
still want a solution that’s as good as possible, even if it’s not the exact, or
optimal, answer. For example, in the Independent Set Problem, even if we can’t
find the largest independent set in a graph, it’s still natural to want to compute
for as much time as we have available, and output as large an independent set
as we can find.
The next few topics in the book will be focused on different aspects
of this notion. In Chapters 11 and 12, we’ll look at algo~-ithms that provide
approximate answers with guaranteed error bounds in polynomial time; we’ll
also consider local search heuristics that are often very effective in practice,
~ We use the term NP-hard to mean "at least as hard as an NP-complete problem." We avoid referring to
optimization problems as NP-complete, since technically this term applies only to decision problems.