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

War Story: Hard Against the Clock
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Proving
War Story: Hard Against the Clock
My class's attention span was running down like sand through an hourglass. Eyes were starting to glaze
even in the front row. Breathing had become soft and regular in the middle of the room. Heads were
tilted back and eyes shut in the rear.
There were fifteen minutes left to go in my lecture on NP-completeness, and I couldn't really blame
them. They had already seen several reductions like the ones presented here, but NP-completeness
reductions are easier to create than to understand or explain. They had to watch one being created in
order to appreciate this.
I reached for my trusty copy of Garey and Johnson's book [GJ79], which contains a list of over four
hundred different known NP-complete problems in an appendix in the back.
``Enough of this!'' I announced loudly enough to startle those in the back row. ``NP-completeness proofs
are routine enough that we can construct them on demand. I need a volunteer with a finger. Can anyone
help me?''
A few students in the front held up their hands. A few students in the back held up their fingers. I opted
for one from the front row.
``Select a problem at random from the back of this book. I can prove the hardness of any of these
problems in the now twelve minutes remaining in this class. Stick your finger in and read me a problem.''
I had definitely gotten their attention. But I could have done that by offering to juggle chainsaws. Now I
had to deliver results without cutting myself into ribbons.
The student picked out a problem. ``OK, prove that Inequivalence of Programs with Assignments is
hard,'' she said.
``Huh? I've never heard of that problem before. What is it? Read me the entire description of the problem
so I can write it on the board.'' The problem was as follows:
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