The.Algorithm.Design.Manual.Springer-Verlag.1998

War Story: Hard Against the Clock
It took me a few minutes worth of scratching before I found the right program. I assumed that I had
access to constants for and , which seemed reasonable, in the sense that it shouldn't make the
problem algorithmically harder. Once my proof worked, I could later worry about removing the extra
assumption if desired.
``Great. Now I have a way to evaluate the truth of each clause. I can do the same thing to evaluate
whether all the clauses are satisfied.''
Now the back of the classroom was getting excited. They were starting to see a ray of hope that they
would get to leave on time. There were two minutes left in class.
``Great. So now we have a program that can evaluate to be true if and only if there is a way to assign the
variables so as to satisfy the set of clauses. We need a second program to finish the job. What about
? Yes, that is all we need. Our language problem asks whether the two programs always
output the same thing, regardless of the possible variable assignments. If the clauses are satisfiable, that
means that there must be an assignment of the variables such that the long program would output true.
Testing whether the programs are equivalent is exactly the same as asking if the clauses are satisfiable.''
I lifted my arms in triumph. ``And so, the problem is neat, sweet, and NP-complete.'' I got the last word
out just before the bell rang.
This exercise was so much fun that I repeated it the next time I taught the course, using a different
randomly selected problem. The audio from my ultimately successful attempt to prove hardness is
accessible from the CD-ROM, if you want to hear what the creation of an NP-completeness proof sounds
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