Tayzwi

Should be reading more and writing less, but well...

Thursday, May 03, 2007

 

Counter Example

Finding counter examples to conjectures can be notoriously hard (pun? I think not). This is an area of creativity that mostly goes unappreciated.

Here's a personal anecdote: my father once came up with an algorithm to solve a hugely constrained version of the traveling salesman problem. The greedy proof was slightly hand-wavy, and I felt it would be an easier thing to find a counter example where his algorithm wouldn't find the optimal tour. Of course, I was just trying to tell him that he couldn't have solved TSP (or even approximated it). I learnt two lessons that day.

Lesson One: One must always speak sweet, because one underestimates the number of times one has to eat his own words.

Lesson Two: Finding counter examples can get quite tricky - and if I may, I would admit that it's not just tricky, it's quite hard - requires tons of patience, and a deep understanding of the problem and the algorithm we are out to disprove. I had learnt a similar lesson earlier in Sundar's Approximation Algorithms class. Sundar let us spend one hour counter-exampling that a minimum spanning tree over the vertex set wouldn't give us a minimum Steiner tree. The counter example used a 'construct' that was quite simple, and took a few minutes of dedicated thought to find. But what amazed me today was this fact about Tait's conjecture:
Tait's conjecture states that "Every polyhedron has a Hamiltonian cycle (along the edges) through all its vertices". It was proposed in 1886 by P. G. Tait and disproved in 1946, when W. T. Tutte constructed a counterexample with 25 faces, 69 edges and 46 vertices.
Imagine the patience, creativity, deep understanding of the problem, and the [least appreciated of them all] ability to borrow from related problems and areas - it takes to come up with such a counter example. And I keep wondering: why research?

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