105 research outputs found
Online Matching with Stochastic Rewards: Optimal Competitive Ratio via Path Based Formulation
The problem of online matching with stochastic rewards is a generalization of
the online bipartite matching problem where each edge has a probability of
success. When a match is made it succeeds with the probability of the
corresponding edge. Introducing this model, Mehta and Panigrahi (FOCS 2012)
focused on the special case of identical edge probabilities. Comparing against
a deterministic offline LP, they showed that the Ranking algorithm of Karp et
al. (STOC 1990) is 0.534 competitive and proposed a new online algorithm with
an improved guarantee of for vanishingly small probabilities. For the
case of vanishingly small but heterogeneous probabilities Mehta et al. (SODA
2015), gave a 0.534 competitive algorithm against the same LP benchmark. For
the more general vertex-weighted version of the problem, to the best of our
knowledge, no results being were previously known even for identical
probabilities.
We focus on the vertex-weighted version and give two improvements. First, we
show that a natural generalization of the Perturbed-Greedy algorithm of
Aggarwal et al. (SODA 2011), is competitive when probabilities
decompose as a product of two factors, one corresponding to each vertex of the
edge. This is the best achievable guarantee as it includes the case of
identical probabilities and in particular, the classical online bipartite
matching problem. Second, we give a deterministic competitive algorithm
for the previously well studied case of fully heterogeneous but vanishingly
small edge probabilities. A key contribution of our approach is the use of
novel path-based analysis. This allows us to compare against the natural
benchmarks of adaptive offline algorithms that know the sequence of arrivals
and the edge probabilities in advance, but not the outcomes of potential
matches.Comment: Preliminary version in EC 202
On the approximability of adjustable robust convex optimization under uncertainty
In this paper, we consider adjustable robust versions of convex optimization problems with uncertain constraints and objectives and show that under fairly general assumptions, a static robust solution provides a good approximation for these adjustable robust problems. An adjustable robust optimization problem is usually intractable since it requires to compute a solution for all possible realizations of uncertain parameters, while an optimal static solution can be computed efficiently in most cases if the corresponding deterministic problem is tractable. The performance of the optimal static robust solution is related to a fundamental geometric property, namely, the symmetry of the uncertainty set. Our work allows for the constraint and objective function coefficients to be uncertain and for the constraints and objective functions to be convex, thereby providing significant extensions of the results in Bertsimas and Goyal (Math Oper Res 35:284β305, 2010) and Bertsimas et al. (Math Oper Res 36: 24β54, 2011b) where only linear objective and linear constraints were considered. The models in this paper encompass a wide variety of problems in revenue management, resource allocation under uncertainty, scheduling problems with uncertain processing times, semidefinite optimization among many others. To the best of our knowledge, these are the first approximation bounds for adjustable robust convex optimization problems in such generality.National Science Foundation (U.S.) (NSF Grant CMMI-1201116
Online Allocation of Reusable Resources via Algorithms Guided by Fluid Approximations
We consider the problem of online allocation (matching and assortments) of
reusable resources where customers arrive sequentially in an adversarial
fashion and allocated resources are used or rented for a stochastic duration
that is drawn independently from known distributions. Focusing on the case of
large inventory, we give an algorithm that is competitive for general
usage distributions. At the heart of our result is the notion of a relaxed
online algorithm that is only subjected to fluid approximations of the
stochastic elements in the problem. The output of this algorithm serves as a
guide for the final algorithm. This leads to a principled approach for
seamlessly addressing stochastic elements (such as reusability, customer
choice, and combinations thereof) in online resource allocation problems, that
may be useful more broadly
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