29 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
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
Algorithms for Secretary Problems on Graphs and Hypergraphs
We examine several online matching problems, with applications to Internet
advertising reservation systems. Consider an edge-weighted bipartite graph G,
with partite sets L, R. We develop an 8-competitive algorithm for the following
secretary problem: Initially given R, and the size of L, the algorithm receives
the vertices of L sequentially, in a random order. When a vertex l \in L is
seen, all edges incident to l are revealed, together with their weights. The
algorithm must immediately either match l to an available vertex of R, or
decide that l will remain unmatched.
Dimitrov and Plaxton show a 16-competitive algorithm for the transversal
matroid secretary problem, which is the special case with weights on vertices,
not edges. (Equivalently, one may assume that for each l \in L, the weights on
all edges incident to l are identical.) We use a similar algorithm, but
simplify and improve the analysis to obtain a better competitive ratio for the
more general problem. Perhaps of more interest is the fact that our analysis is
easily extended to obtain competitive algorithms for similar problems, such as
to find disjoint sets of edges in hypergraphs where edges arrive online. We
also introduce secretary problems with adversarially chosen groups. Finally, we
give a 2e-competitive algorithm for the secretary problem on graphic matroids,
where, with edges appearing online, the goal is to find a maximum-weight
acyclic subgraph of a given graph.Comment: 15 pages, 2 figure