62 research outputs found
Normal all pseudo-Anosov subgroups of mapping class groups
We construct the first known examples of nontrivial, normal, all
pseudo-Anosov subgroups of mapping class groups of surfaces. Specifically, we
construct such subgroups for the closed genus two surface and for the sphere
with five or more punctures. Using the branched covering of the genus two
surface over the sphere and results of Birman and Hilden, we prove that a
reducible mapping class of the genus two surface projects to a reducible
mapping class on the sphere with six punctures. The construction introduces
"Brunnian" mapping classes of the sphere, which are analogous to Brunnian
links.Comment: Published in Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol4/paper10.abs.htm
Minimum-Cost Coverage of Point Sets by Disks
We consider a class of geometric facility location problems in which the goal
is to determine a set X of disks given by their centers (t_j) and radii (r_j)
that cover a given set of demand points Y in the plane at the smallest possible
cost. We consider cost functions of the form sum_j f(r_j), where f(r)=r^alpha
is the cost of transmission to radius r. Special cases arise for alpha=1 (sum
of radii) and alpha=2 (total area); power consumption models in wireless
network design often use an exponent alpha>2. Different scenarios arise
according to possible restrictions on the transmission centers t_j, which may
be constrained to belong to a given discrete set or to lie on a line, etc. We
obtain several new results, including (a) exact and approximation algorithms
for selecting transmission points t_j on a given line in order to cover demand
points Y in the plane; (b) approximation algorithms (and an algebraic
intractability result) for selecting an optimal line on which to place
transmission points to cover Y; (c) a proof of NP-hardness for a discrete set
of transmission points in the plane and any fixed alpha>1; and (d) a
polynomial-time approximation scheme for the problem of computing a minimum
cost covering tour (MCCT), in which the total cost is a linear combination of
the transmission cost for the set of disks and the length of a tour/path that
connects the centers of the disks.Comment: 10 pages, 4 figures, Latex, to appear in ACM Symposium on
Computational Geometry 200
Greedy Optimal Homotopy and Homology Generators
We describe simple greedy algorithms to construct the shortest set of loops that generates either the fundamental group (with a given basepoint) or the first homology group (over any fixed coefficient field) of any oriented 2-manifold. In particular, we show that the shortest set of loops that generate the fundamental group of any oriented combinatorial 2-manifold, with any given basepoint, can be constructed in O(n log n) time using a straightforward application of Dijkstra’s shortest path algorithm. This solves an open problem of Colin de Verdière and Lazarus
- …