9 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
Periodic Reranking for Online Matching of Reusable Resources
We consider a generalization of the vertex weighted online bipartite matching
problem where the offline vertices, called resources, are reusable. In
particular, when a resource is matched it is unavailable for a deterministic
time duration after which it becomes available for a re-match. Thus, a
resource can be matched to many different online vertices over a period of
time. While recent work on the problem has resolved the asymptotic case where
we have large starting inventory (i.e., many copies) of every resource, we
consider the (more general) case of unit inventory and give the first algorithm
that is provably better than the na\"ive greedy approach which has a
competitive ratio of (exactly) 0.5. In particular, we achieve a competitive
ratio of 0.589 against an LP relaxation of the offline problem.Comment: ACM EC 202
Robust Appointment Scheduling with Heterogeneous Costs
Designing simple appointment systems that under uncertainty in service times, try to achieve both high utilization of expensive medical equipment and personnel as well as short waiting time for patients, has long been an interesting and challenging problem in health care. We consider a robust version of the appointment scheduling problem, introduced by Mittal et al. (2014), with the goal of finding simple and easy-to-use algorithms. Previous work focused on the special case where per-unit costs due to under-utilization of equipment/personnel are homogeneous i.e., costs are linear and identical. We consider the heterogeneous case and devise an LP that has a simple closed-form solution. This solution yields the first constant-factor approximation for the problem. We also find special cases beyond homogeneous costs where the LP leads to closed form optimal schedules. Our approach and results extend more generally to convex piece-wise linear costs.
For the case where the order of patients is changeable, we focus on linear costs and show that the problem is strongly NP-hard when the under-utilization costs are heterogeneous. For changeable order with homogeneous under-utilization costs, it was previously shown that an EPTAS exists. We instead find an extremely simple, ratio-based ordering that is 1.0604 approximate
Vignettes on robust combinatorial optimization
Thesis: Ph. D., Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2018.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (pages 137-142).In this thesis, we design and analyze algorithms for robust combinatorial optimization in various settings. First, we consider the problem of simultaneously maximizing multiple objectives, all monotone submodular, subject to a cardinality constraint. We focus on the case where the number of objectives is super-constant yet much smaller than the cardinality of the chosen set. We propose several algorithms (including one with the best achievable asymptotic guarantee for the problem). Experiments on synthetic data show that a heuristic based on our more practical and fast algorithm outperforms current practical algorithms in all cases considered. Next, we study the problem of robust maximization of a single monotone submodular function in scenarios where after choosing a feasible set of elements, some elements from the chosen set are adversarially removed. Under some restriction on the number of elements that can be removed, we give the first constant factor approximation algorithms as well as the best possible asymptotic approximation in certain cases. We also give a black box result for the much more general setting of deletion-robust maximization subject to an independence system. Lastly, we consider a robust appointment scheduling problem where the goal is to design simple appointment systems that try to achieve both high server utilization as well as short waiting times, under uncertainty in job processing times. When the order of jobs is fixed and one seeks to find optimal appointment duration for jobs, we give a simple heuristic that achieves the first constant factor (2) approximation. We also give closed form optimal solutions in various special cases that supersede previous work. For the setting where order of jobs is also flexible and under-utilization costs are homogeneous, it was previously shown that an EPTAS exists. We instead focus on simple and practical heuristics, and find a ratio based ordering which is 1.0604 approximate, improving on previous results for similarly practical heuristics.by Rajan Udwani.Ph. D
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
Multi-objective Maximization of Monotone Submodular Functions with Cardinality Constraint
We consider the problem of multi-objective maximization of monotone submodular functions subject to cardinality constraint, often formulated as . While it is widely known that greedy methods work well for a single objective, the problem becomes much harder with multiple objectives. In fact, Krause et al.\ (2008) showed that when the number of objectives grows as the cardinality i.e., , the problem is inapproximable (unless ). We focus on the case where the number of objectives is super-constant yet much smaller than the cardinality of the chosen set. We propose the first constant factor algorithms for this regime, including one with the best achievable asymptotic guarantee and also a more practical nearly linear time algorithm. Experiments on synthetic data show that a heuristic based on our more practical and fast algorithm outperforms existing heuristics.Non UBCUnreviewedAuthor affiliation: Columbia UniversityPostdoctora
Robust Monotone Submodular Function Maximization
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9682)We consider a robust formulation, introduced by Krause et al. (2008), of the classic cardinality constrained monotone submodular function maximization problem, and give the first constant factor approximation results. The robustness considered is w.r.t. adversarial removal of a given number of elements from the chosen set. In particular, for the fundamental case of single element removal, we show that one can approximate the problem up to a factor (1−1/e)−ϵ by making O(n 1/ϵ) queries, for arbitrary ϵ > 0. The ideas are also extended to more general settings