Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Prof. Emo Welzl and Prof. Bernd Gärtner

Mittagsseminar Talk Information |

**Date and Time**: Tuesday, October 06, 2015, 12:15 pm

**Duration**: 30 minutes

**Location**: OAT S15/S16/S17

**Speaker**: Emo Welzl

Consider a planar finite point set *P*, no three on a line and exactly one point not extreme in *P*. We call this a *wheel configuration* and we are interested in pm(*P*), the number of crossing-free perfect matchings on *P*. (If, contrary to our assumption, all points in a set *S* are extreme, i.e. in convex position, then it is well-known that pm(*S*) = *C _{m}*, the

We give exact tight upper and lower bounds on pm(*P*) dependent on |*P*|. Simplified to its asymptotics in terms of *C _{m}*, these yield

(9/8) *C _{m}* (1 - o(1)) $\le$ pm(P) $\le$ (3/2)

We characterize the sets (order types) which maximize or minimize pm(*P*). Moreover, among all sets *S* of a given size not in convex position, pm(*S*) is minimized for some wheel configuration (this follows along a short excursion to well formed parentheses strings). Therefore, leaving convex position increases the number of crossing-free perfect matchings by at least a factor of 9/8 (in the limit as |*S*| grows). From what we show it follows that pm(*P*) can be computed efficiently.

A connection to related problems (triangulations, origin embracing triangles) is briefly discussed. (Joint work with Andres J. Ruiz-Vargas, EPFL.)

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