Perfect matchings with a high stabbing number

Once upon a time, an idiosyncratic king set up a peculiar system for settling marriages in his kingdom. Once every year, he would invite all the couples that wished to tie the knot to a grand ceremony. Upon arrival, the couples would be taken to a large open area with chairs that were fixed to the ground spread all around and asked to get seated. The arrangements would be so made that the chairs would neither be surplus nor in shortage. Thus each individual would get exactly one chair and no more.

Finally, the king himself would arrive, examine the seated guests, draw one long line on the ground and stand on one of its two sides. That’s when the marriages would be decided. Couples where both partners were seated on the side of the line the king stood on would get married and couples separated by the holy line would be forbidden from seeing each other ever again.

The king wanted to slow down the recent exponential growth in population in his kingdom and so he wanted as few couples to be married as possible. Since he had complete knowledge of who wanted to get married to whom, he could, in principle, devise an evil arrangement of chairs and draw one really mean line that would separate most of the couples at the ceremony. On the other hand, the couples were allowed to collude with each other upon seeing the arrangement of chairs and decide who got to sit where. Thus perhaps they could formulate a clever strategy that would let most of them be on the same side as the king no matter what line he chose to draw?

Year after year passed by and the king, drawing upon the wisdom of the entire royal ministry, managed to hoodwink his people and successfully stalled most of the romance in his kingdom. The lack of expertise in computational geometry among the general public proved to be detrimental to them. The grand ceremony, having the flavor of a gripping puzzle, got the king addicted and very soon, by developing progressively sophisticated and elaborate strategies, he unknowingly brought his own kingdom to what could be described as extinction.

Centuries later, in the year 1989, two researchers, trying to design an efficient data structure to perform range searching queries on a point-set proved an interesting theorem. They weren’t aware that the theorem held the key to a centuries old conundrum that could have saved an entire kingdom from going extinct. What they proved essentially amounted to this:

“No matter what the arrangement of chairs, the couples can always collude with each other and compute an assignment of chairs to each individual, so that no matter what line the king draws and no matter what side he stands on, at least a polynomial number of them get married.”

In fact, they proved something even stronger. Their theorem does not so much depend on the fact that the shape the king draws is a line. Other geometric shapes, such as circles, rectangles, squares, triangles, can all be plugged into the theorem in place of “line” and the statement will still hold true.

As long as the shape satisfies the property that its dual shatter function is polynomial, the theorem works. The dual shatter function for a shape is the maximum number of cells one can get in a Venn diagram obtained by drawing n of those shapes. For example, for the case of halfplanes (i.e., a line and one of its sides), one can easily show using induction on the number of lines that the dual shatter function is polynomial. Notice that when incrementally adding a halfplane to a partially built Venn diagram of halfplanes, the number of new cells created is equal to the number of cells this new halfplane’s boundary intersects. Since a new line can intersect an old line at most once, the number of cells it intersects is at most the number of lines already present. Thus the dual shatter function is O(n^2). Simiarly, for any shape that satisfies the property that boundaries of two instances of the shape always intersect in a constant number of places, the dual shatter function is bounded from above by a polynomial.

Actually, the theorem does not just hold in the geometric setting. It holds for general set systems. Thus if the ceremony were organized in interstellar space with chairs occupying co-ordinates in three dimensions, or in some bizarre abstract space, a polynomial number of marriages could be saved as long as the shape chosen by the king had a polynomial dual shatter function.

(Bonus points if you can correctly guess the definition of stabbing number without looking it up.)

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