46 research outputs found
Advances on Matroid Secretary Problems: Free Order Model and Laminar Case
The most well-known conjecture in the context of matroid secretary problems
claims the existence of a constant-factor approximation applicable to any
matroid. Whereas this conjecture remains open, modified forms of it were shown
to be true, when assuming that the assignment of weights to the secretaries is
not adversarial but uniformly random (Soto [SODA 2011], Oveis Gharan and
Vondr\'ak [ESA 2011]). However, so far, there was no variant of the matroid
secretary problem with adversarial weight assignment for which a
constant-factor approximation was found. We address this point by presenting a
9-approximation for the \emph{free order model}, a model suggested shortly
after the introduction of the matroid secretary problem, and for which no
constant-factor approximation was known so far. The free order model is a
relaxed version of the original matroid secretary problem, with the only
difference that one can choose the order in which secretaries are interviewed.
Furthermore, we consider the classical matroid secretary problem for the
special case of laminar matroids. Only recently, a constant-factor
approximation has been found for this case, using a clever but rather involved
method and analysis (Im and Wang, [SODA 2011]) that leads to a
16000/3-approximation. This is arguably the most involved special case of the
matroid secretary problem for which a constant-factor approximation is known.
We present a considerably simpler and stronger -approximation, based on reducing the problem to a matroid secretary
problem on a partition matroid
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
Laminar Matroids
A laminar family is a collection of subsets of a set such
that, for any two intersecting sets, one is contained in the other. For a
capacity function on , let be \{I:|I\cap A|
\leq c(A)\text{ for all A\in\mathscr{A}}\}. Then is the
collection of independent sets of a (laminar) matroid on . We present a
method of compacting laminar presentations, characterize the class of laminar
matroids by their excluded minors, present a way to construct all laminar
matroids using basic operations, and compare the class of laminar matroids to
other well-known classes of matroids.Comment: 17 page
The Best-or-Worst and the Postdoc problems
We consider two variants of the secretary problem, the\emph{ Best-or-Worst}
and the \emph{Postdoc} problems, which are closely related. First, we prove
that both variants, in their standard form with binary payoff 1 or 0, share the
same optimal stopping rule. We also consider additional cost/perquisites
depending on the number of interviewed candidates. In these situations the
optimal strategies are very different. Finally, we also focus on the
Best-or-Worst variant with different payments depending on whether the selected
candidate is the best or the worst
Generalized Laminar Matroids
Nested matroids were introduced by Crapo in 1965 and have appeared frequently
in the literature since then. A flat of a matroid is Hamiltonian if it has
a spanning circuit. A matroid is nested if and only if its Hamiltonian
flats form a chain under inclusion; is laminar if and only if, for every
-element independent set , the Hamiltonian flats of containing
form a chain under inclusion. We generalize these notions to define the classes
of -closure-laminar and -laminar matroids. This paper focuses on
structural properties of these classes noting that, while the second class is
always minor-closed, the first is if and only if . The main results
are excluded-minor characterizations for the classes of 2-laminar and
2-closure-laminar matroids.Comment: 12 page