7 research outputs found
The matroid secretary problem for minor-closed classes and random matroids
We prove that for every proper minor-closed class of matroids
representable over a prime field, there exists a constant-competitive matroid
secretary algorithm for the matroids in . This result relies on the
extremely powerful matroid minor structure theory being developed by Geelen,
Gerards and Whittle.
We also note that for asymptotically almost all matroids, the matroid
secretary algorithm that selects a random basis, ignoring weights, is
-competitive. In fact, assuming the conjecture that almost all
matroids are paving, there is a -competitive algorithm for almost all
matroids.Comment: 15 pages, 0 figure
On the density of matroids omitting a complete-graphic minor
We show that, if is a simple rank- matroid with no -point line
minor and no minor isomorphic to the cycle matroid of a -vertex complete
graph, then the ratio is bounded above by a singly exponential
function of and . We also bound this ratio in the special case where
is a frame matroid, obtaining an answer that is within a factor of two of
best-possible.Comment: 25 page
Clique minors in dense matroids
The objective of this thesis is to bound the number of points a - and -minor-free matroid has. We first prove that a sufficiently large matroid will contain a structure called a tower. We then use towers to find a complete minor in a matroid with no -minor
Non-Adaptive Matroid Prophet Inequalities
We consider the problem of matroid prophet inequalities. This problem has been ex-
tensively studied in case of adaptive prices, with [KW12] obtaining a tight 2-competitive
mechanism for all the matroids.
However, the case non-adaptive is far from resolved, although there is a known constant-
competitive mechanism for uniform and graphical matroids (see [Cha+20]).
We improve on constant-competitive mechanism from [Cha+20] for graphical matroids,
present a separate mechanism for cographical matroids, and combine those to obtain
constant-competitive mechanism for all regular matroids