188 research outputs found
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
Online Independent Set Beyond the Worst-Case: Secretaries, Prophets, and Periods
We investigate online algorithms for maximum (weight) independent set on
graph classes with bounded inductive independence number like, e.g., interval
and disk graphs with applications to, e.g., task scheduling and spectrum
allocation. In the online setting, it is assumed that nodes of an unknown graph
arrive one by one over time. An online algorithm has to decide whether an
arriving node should be included into the independent set. Unfortunately, this
natural and practically relevant online problem cannot be studied in a
meaningful way within a classical competitive analysis as the competitive ratio
on worst-case input sequences is lower bounded by .
As a worst-case analysis is pointless, we study online independent set in a
stochastic analysis. Instead of focussing on a particular stochastic input
model, we present a generic sampling approach that enables us to devise online
algorithms achieving performance guarantees for a variety of input models. In
particular, our analysis covers stochastic input models like the secretary
model, in which an adversarial graph is presented in random order, and the
prophet-inequality model, in which a randomly generated graph is presented in
adversarial order. Our sampling approach bridges thus between stochastic input
models of quite different nature. In addition, we show that our approach can be
applied to a practically motivated admission control setting.
Our sampling approach yields an online algorithm for maximum independent set
with competitive ratio with respect to all of the mentioned
stochastic input models. for graph classes with inductive independence number
. The approach can be extended towards maximum-weight independent set by
losing only a factor of in the competitive ratio with denoting
the (expected) number of nodes
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
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
Online Knapsack Problem under Expected Capacity Constraint
Online knapsack problem is considered, where items arrive in a sequential
fashion that have two attributes; value and weight. Each arriving item has to
be accepted or rejected on its arrival irrevocably. The objective is to
maximize the sum of the value of the accepted items such that the sum of their
weights is below a budget/capacity. Conventionally a hard budget/capacity
constraint is considered, for which variety of results are available. In modern
applications, e.g., in wireless networks, data centres, cloud computing, etc.,
enforcing the capacity constraint in expectation is sufficient. With this
motivation, we consider the knapsack problem with an expected capacity
constraint. For the special case of knapsack problem, called the secretary
problem, where the weight of each item is unity, we propose an algorithm whose
probability of selecting any one of the optimal items is equal to and
provide a matching lower bound. For the general knapsack problem, we propose an
algorithm whose competitive ratio is shown to be that is significantly
better than the best known competitive ratio of for the knapsack
problem with the hard capacity constraint.Comment: To appear in IEEE INFOCOM 2018, April 2018, Honolulu H
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