24 research outputs found
Approximation algorithms for stochastic and risk-averse optimization
We present improved approximation algorithms in stochastic optimization. We
prove that the multi-stage stochastic versions of covering integer programs
(such as set cover and vertex cover) admit essentially the same approximation
algorithms as their standard (non-stochastic) counterparts; this improves upon
work of Swamy \& Shmoys which shows an approximability that depends
multiplicatively on the number of stages. We also present approximation
algorithms for facility location and some of its variants in the -stage
recourse model, improving on previous approximation guarantees. We give a
-approximation algorithm in the standard polynomial-scenario model and
an algorithm with an expected per-scenario -approximation guarantee,
which is applicable to the more general black-box distribution model.Comment: Extension of a SODA'07 paper. To appear in SIAM J. Discrete Mat
Matching with Commitments
We consider the following stochastic optimization problem first introduced by
Chen et al. in \cite{chen}. We are given a vertex set of a random graph where
each possible edge is present with probability p_e. We do not know which edges
are actually present unless we scan/probe an edge. However whenever we probe an
edge and find it to be present, we are constrained to picking the edge and both
its end points are deleted from the graph. We wish to find the maximum matching
in this model. We compare our results against the optimal omniscient algorithm
that knows the edges of the graph and present a 0.573 factor algorithm using a
novel sampling technique. We also prove that no algorithm can attain a factor
better than 0.898 in this model
Stochastic Vehicle Routing with Recourse
We study the classic Vehicle Routing Problem in the setting of stochastic
optimization with recourse. StochVRP is a two-stage optimization problem, where
demand is satisfied using two routes: fixed and recourse. The fixed route is
computed using only a demand distribution. Then after observing the demand
instantiations, a recourse route is computed -- but costs here become more
expensive by a factor lambda.
We present an O(log^2 n log(n lambda))-approximation algorithm for this
stochastic routing problem, under arbitrary distributions. The main idea in
this result is relating StochVRP to a special case of submodular orienteering,
called knapsack rank-function orienteering. We also give a better approximation
ratio for knapsack rank-function orienteering than what follows from prior
work. Finally, we provide a Unique Games Conjecture based omega(1) hardness of
approximation for StochVRP, even on star-like metrics on which our algorithm
achieves a logarithmic approximation.Comment: 20 Pages, 1 figure Revision corrects the statement and proof of
Theorem 1.
Adaptive Seeding in Social Networks
The algorithmic challenge of maximizing information diffusion through word-of-mouth processes in social networks has been heavily studied in the past decade. While there has been immense progress and an impressive arsenal of techniques has been developed, the algorithmic frameworks make idealized assumptions regarding access to the network that can often result in poor performance of state-of-the-art techniques. In this paper we introduce a new framework which we call Adaptive Seeding. The framework is a two-stage stochastic optimization model designed to leverage the potential that typically lies in neighboring nodes of arbitrary samples of social networks. Our main result is an algorithm which provides a constant factor approximation to the optimal adaptive policy for any influence function in the Triggering model
Approximation Algorithms for the Stochastic Lot-Sizing Problem with Order Lead Times
We develop new algorithmic approaches to compute provably near-optimal policies for multiperiod stochastic lot-sizing inventory models with positive lead times, general demand distributions, and dynamic forecast updates. The policies that are developed have worst-case performance guarantees of 3 and typically perform very close to optimal in extensive computational experiments. The newly proposed algorithms employ a novel randomized decision rule. We believe that these new algorithmic and performance analysis techniques could be used in designing provably near-optimal randomized algorithms for other stochastic inventory control models and more generally in other multistage stochastic control problems.National Science Foundation (U.S.) (Grant DMS-0732175)National Science Foundation (U.S.) (CAREER Award CMMI-0846554)United States. Air Force Office of Scientific Research (Award FA9550-08-1-0369)United States. Air Force Office of Scientific Research (Award FA9550-11-1-0150)Singapore-MIT AllianceSolomon Buchsbaum AT&T Research Fun
Set covering with our eyes closed
Given a universe of elements and a weighted collection of subsets of , the universal set cover problem is to a priori map each element to a set containing such that any set is covered by S(X)=\cup_{u\in XS(u). The aim is to find a mapping such that the cost of is as close as possible to the optimal set cover cost for . (Such problems are also called oblivious or a priori optimization problems.) Unfortunately, for every universal mapping, the cost of can be times larger than optimal if the set is adversarially chosen. In this paper we study the performance on average, when is a set of randomly chosen elements from the universe: we show how to efficiently find a universal map whose expected cost is times the expected optimal cost. In fact, we give a slightly improved analysis and show that this is the best possible. We generalize these ideas to weighted set cover and show similar guarantees to (nonmetric) facility location, where we have to balance the facility opening cost with the cost of connecting clients to the facilities. We show applications of our results to universal multicut and disc-covering problems and show how all these universal mappings give us algorithms for the stochastic online variants of the problems with the same competitive factors