2,308 research outputs found
The Submodular Secretary Problem Goes Linear
During the last decade, the matroid secretary problem (MSP) became one of the
most prominent classes of online selection problems. Partially linked to its
numerous applications in mechanism design, substantial interest arose also in
the study of nonlinear versions of MSP, with a focus on the submodular matroid
secretary problem (SMSP). So far, O(1)-competitive algorithms have been
obtained for SMSP over some basic matroid classes. This created some hope that,
analogously to the matroid secretary conjecture, one may even obtain
O(1)-competitive algorithms for SMSP over any matroid. However, up to now, most
questions related to SMSP remained open, including whether SMSP may be
substantially more difficult than MSP; and more generally, to what extend MSP
and SMSP are related.
Our goal is to address these points by presenting general black-box
reductions from SMSP to MSP. In particular, we show that any O(1)-competitive
algorithm for MSP, even restricted to a particular matroid class, can be
transformed in a black-box way to an O(1)-competitive algorithm for SMSP over
the same matroid class. This implies that the matroid secretary conjecture is
equivalent to the same conjecture for SMSP. Hence, in this sense SMSP is not
harder than MSP. Also, to find O(1)-competitive algorithms for SMSP over a
particular matroid class, it suffices to consider MSP over the same matroid
class. Using our reductions we obtain many first and improved O(1)-competitive
algorithms for SMSP over various matroid classes by leveraging known algorithms
for MSP. Moreover, our reductions imply an O(loglog(rank))-competitive
algorithm for SMSP, thus, matching the currently best asymptotic algorithm for
MSP, and substantially improving on the previously best
O(log(rank))-competitive algorithm for SMSP
Buyback Problem - Approximate matroid intersection with cancellation costs
In the buyback problem, an algorithm observes a sequence of bids and must
decide whether to accept each bid at the moment it arrives, subject to some
constraints on the set of accepted bids. Decisions to reject bids are
irrevocable, whereas decisions to accept bids may be canceled at a cost that is
a fixed fraction of the bid value. Previous to our work, deterministic and
randomized algorithms were known when the constraint is a matroid constraint.
We extend this and give a deterministic algorithm for the case when the
constraint is an intersection of matroid constraints. We further prove a
matching lower bound on the competitive ratio for this problem and extend our
results to arbitrary downward closed set systems. This problem has applications
to banner advertisement, semi-streaming, routing, load balancing and other
problems where preemption or cancellation of previous allocations is allowed
Matroid Online Bipartite Matching and Vertex Cover
The Adwords and Online Bipartite Matching problems have enjoyed a renewed
attention over the past decade due to their connection to Internet advertising.
Our community has contributed, among other things, new models (notably
stochastic) and extensions to the classical formulations to address the issues
that arise from practical needs. In this paper, we propose a new generalization
based on matroids and show that many of the previous results extend to this
more general setting. Because of the rich structures and expressive power of
matroids, our new setting is potentially of interest both in theory and in
practice.
In the classical version of the problem, the offline side of a bipartite
graph is known initially while vertices from the online side arrive one at a
time along with their incident edges. The objective is to maintain a decent
approximate matching from which no edge can be removed. Our generalization,
called Matroid Online Bipartite Matching, additionally requires that the set of
matched offline vertices be independent in a given matroid. In particular, the
case of partition matroids corresponds to the natural scenario where each
advertiser manages multiple ads with a fixed total budget.
Our algorithms attain the same performance as the classical version of the
problems considered, which are often provably the best possible. We present
-competitive algorithms for Matroid Online Bipartite Matching under the
small bid assumption, as well as a -competitive algorithm for Matroid
Online Bipartite Matching in the random arrival model. A key technical
ingredient of our results is a carefully designed primal-dual waterfilling
procedure that accommodates for matroid constraints. This is inspired by the
extension of our recent charging scheme for Online Bipartite Vertex Cover.Comment: 19 pages, to appear in EC'1
The Maximum Likelihood Threshold of a Graph
The maximum likelihood threshold of a graph is the smallest number of data
points that guarantees that maximum likelihood estimates exist almost surely in
the Gaussian graphical model associated to the graph. We show that this graph
parameter is connected to the theory of combinatorial rigidity. In particular,
if the edge set of a graph is an independent set in the -dimensional
generic rigidity matroid, then the maximum likelihood threshold of is less
than or equal to . This connection allows us to prove many results about the
maximum likelihood threshold.Comment: Added Section 6 and Section
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