18 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
Optimal Composition Ordering Problems for Piecewise Linear Functions
In this paper, we introduce maximum composition ordering problems. The input
is real functions and a constant
. We consider two settings: total and partial compositions. The
maximum total composition ordering problem is to compute a permutation
which maximizes , where .
The maximum partial composition ordering problem is to compute a permutation
and a nonnegative integer which maximize
.
We propose time algorithms for the maximum total and partial
composition ordering problems for monotone linear functions , which
generalize linear deterioration and shortening models for the time-dependent
scheduling problem. We also show that the maximum partial composition ordering
problem can be solved in polynomial time if is of form
for some constants , and . We
finally prove that there exists no constant-factor approximation algorithm for
the problems, even if 's are monotone, piecewise linear functions with at
most two pieces, unless P=NP.Comment: 19 pages, 4 figure
Prophet Inequalities with Limited Information
In the classical prophet inequality, a gambler observes a sequence of
stochastic rewards and must decide, for each reward ,
whether to keep it and stop the game or to forfeit the reward forever and
reveal the next value . The gambler's goal is to obtain a constant
fraction of the expected reward that the optimal offline algorithm would get.
Recently, prophet inequalities have been generalized to settings where the
gambler can choose items, and, more generally, where he can choose any
independent set in a matroid. However, all the existing algorithms require the
gambler to know the distribution from which the rewards are
drawn.
The assumption that the gambler knows the distribution from which
are drawn is very strong. Instead, we work with the much simpler
assumption that the gambler only knows a few samples from this distribution. We
construct the first single-sample prophet inequalities for many settings of
interest, whose guarantees all match the best possible asymptotically,
\emph{even with full knowledge of the distribution}. Specifically, we provide a
novel single-sample algorithm when the gambler can choose any elements
whose analysis is based on random walks with limited correlation. In addition,
we provide a black-box method for converting specific types of solutions to the
related \emph{secretary problem} to single-sample prophet inequalities, and
apply it to several existing algorithms. Finally, we provide a constant-sample
prophet inequality for constant-degree bipartite matchings.
We apply these results to design the first posted-price and multi-dimensional
auction mechanisms with limited information in settings with asymmetric
bidders
The Simulated Greedy Algorithm for Several Submodular Matroid Secretary Problems
We study the matroid secretary problems with submodular valuation functions.
In these problems, the elements arrive in random order. When one element
arrives, we have to make an immediate and irrevocable decision on whether to
accept it or not. The set of accepted elements must form an {\em independent
set} in a predefined matroid. Our objective is to maximize the value of the
accepted elements. In this paper, we focus on the case that the valuation
function is a non-negative and monotonically non-decreasing submodular
function.
We introduce a general algorithm for such {\em submodular matroid secretary
problems}. In particular, we obtain constant competitive algorithms for the
cases of laminar matroids and transversal matroids. Our algorithms can be
further applied to any independent set system defined by the intersection of a
{\em constant} number of laminar matroids, while still achieving constant
competitive ratios. Notice that laminar matroids generalize uniform matroids
and partition matroids.
On the other hand, when the underlying valuation function is linear, our
algorithm achieves a competitive ratio of 9.6 for laminar matroids, which
significantly improves the previous result.Comment: preliminary version appeared in STACS 201
Matroid prophet inequalities and Bayesian mechanism design
Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 42-44).Consider a gambler who observes a sequence of independent, non-negative random numbers and is allowed to stop the sequence at any time, claiming a reward equal to the most recent observation. The famous prophet inequality of Krengel, Sucheston, and Garling asserts that a gambler who knows the distribution of each random variable can achieve at least half as much reward, in expectation, as a "prophet" who knows the sampled values of each random variable and can choose the largest one. We generalize this result to the setting in which the gambler and the prophet are allowed to make more than one selection, subject to a matroid constraint. We show that the gambler can still achieve at least half as much reward as the prophet; this result is the best possible, since it is known that the ratio cannot be improved even in the original prophet inequality, which corresponds to the special case of rank-one matroids. Generalizing the result still further, we show that under an intersection of p matroid constraints, the prophet's reward exceeds the gambler's by a factor of at most 0(p), and this factor is also tight. Beyond their interest as theorems about pure online algoritms or optimal stopping rules, these results also have applications to mechanism design. Our results imply improved bounds on the ability of sequential posted-price mechanisms to approximate optimal mechanisms in both single-parameter and multi-parameter Bayesian settings. In particular, our results imply the first efficiently computable constant-factor approximations to the Bayesian optimal revenue in certain multi-parameter settings. This work was done in collaboration with Robert Kleinberg.by S. Matthew Weinberg.S.M