32 research outputs found
Polymatroid Prophet Inequalities
Consider a gambler and a prophet who observe a sequence of independent,
non-negative numbers. The gambler sees the numbers one-by-one whereas the
prophet sees the entire sequence at once. The goal of both is to decide on
fractions of each number they want to keep so as to maximize the weighted
fractional sum of the numbers chosen.
The classic result of Krengel and Sucheston (1977-78) asserts that if both
the gambler and the prophet can pick one number, then the gambler can do at
least half as well as the prophet. Recently, Kleinberg and Weinberg (2012) have
generalized this result to settings where the numbers that can be chosen are
subject to a matroid constraint.
In this note we go one step further and show that the bound carries over to
settings where the fractions that can be chosen are subject to a polymatroid
constraint. This bound is tight as it is already tight for the simple setting
where the gambler and the prophet can pick only one number. An interesting
application of our result is in mechanism design, where it leads to improved
results for various problems
On Optimality of Myopic Sensing Policy with Imperfect Sensing in Multi-channel Opportunistic Access
We consider the channel access problem under imperfect sensing of channel
state in a multi-channel opportunistic communication system, where the state of
each channel evolves as an independent and identically distributed Markov
process. The considered problem can be cast into a restless multi-armed bandit
(RMAB) problem that is of fundamental importance in decision theory. It is
well-known that solving the RMAB problem is PSPACE-hard, with the optimal
policy usually intractable due to the exponential computation complexity. A
natural alternative is to consider the easily implementable myopic policy that
maximizes the immediate reward but ignores the impact of the current strategy
on the future reward. In this paper, we perform an analytical study on the
optimality of the myopic policy under imperfect sensing for the considered RMAB
problem. Specifically, for a family of generic and practically important
utility functions, we establish the closed-form conditions under which the
myopic policy is guaranteed to be optimal even under imperfect sensing. Despite
our focus on the opportunistic channel access, the obtained results are generic
in nature and are widely applicable in a wide range of engineering domains.Comment: 21 pages regular pape