6 research outputs found
Designing cost-sharing methods for Bayesian games
We study the design of cost-sharing protocols for two fundamental resource allocation problems, the Set Cover and the Steiner Tree Problem, under environments of incomplete information (Bayesian model). Our objective is to design protocols where the worst-case Bayesian Nash equilibria, have low cost, i.e. the Bayesian Price of Anarchy (PoA) is minimized. Although budget balance is a very natural requirement, it puts considerable restrictions on the design space, resulting in high PoA. We propose an alternative, relaxed requirement called budget balance in the equilibrium (BBiE).We show an interesting connection between algorithms for Oblivious Stochastic optimization problems and cost-sharing design with low PoA. We exploit this connection for both problems and we enforce approximate solutions of the stochastic problem, as Bayesian Nash equilibria, with the same guarantees on the PoA. More interestingly, we show how to obtain the same bounds on the PoA, by using anonymous posted prices which are desirable because they are easy to implement and, as we show, induce dominant strategies for the players
Approximation Algorithms for the A Priori TravelingRepairman
We consider the a priori traveling repairman problem, which is a stochastic
version of the classic traveling repairman problem (also called the traveling
deliveryman or minimum latency problem). Given a metric with a root
, the traveling repairman problem (TRP) involves finding a tour
originating from that minimizes the sum of arrival-times at all vertices.
In its a priori version, we are also given independent probabilities of each
vertex being active. We want to find a master tour originating from
and visiting all vertices. The objective is to minimize the expected sum of
arrival-times at all active vertices, when is shortcut over the inactive
vertices. We obtain the first constant-factor approximation algorithm for a
priori TRP under non-uniform probabilities. Previously, such a result was only
known for uniform probabilities
Approximating connected facility location problems via Random facility sampling and core detouring
We present a simple randomized algorithmic framework for connected facility location problems. The basic idea is as follows: We run a black-box approximation algorithm for the unconnected facility location problem, randomly sample the clients, and open the facilities serving sampled clients in the approximate solution. Via a novel analytical tool, which we term core detouring, we show that this approach significantly improves over the previously best known approximation ratios for several NP-hard network design problems. For example, we reduce the approximation ratio for the connected facility location problem from 8.55 to 4.00 and for the single-sink rent-or-buy problem from 3.55 to 2.92. We show that our connected facility location algorithms can be derandomized at the expense of a slightly worse approximation ratio. The versatility of our framework is demonstrated by devising improved approximation algorithms also for other related problems
Designing Networks with Good Equilibria under Uncertainty
We consider the problem of designing network cost-sharing protocols with good
equilibria under uncertainty. The underlying game is a multicast game in a
rooted undirected graph with nonnegative edge costs. A set of k terminal
vertices or players need to establish connectivity with the root. The social
optimum is the Minimum Steiner Tree. We are interested in situations where the
designer has incomplete information about the input. We propose two different
models, the adversarial and the stochastic. In both models, the designer has
prior knowledge of the underlying metric but the requested subset of the
players is not known and is activated either in an adversarial manner
(adversarial model) or is drawn from a known probability distribution
(stochastic model).
In the adversarial model, the designer's goal is to choose a single,
universal protocol that has low Price of Anarchy (PoA) for all possible
requested subsets of players. The main question we address is: to what extent
can prior knowledge of the underlying metric help in the design? We first
demonstrate that there exist graphs (outerplanar) where knowledge of the
underlying metric can dramatically improve the performance of good network
design. Then, in our main technical result, we show that there exist graph
metrics, for which knowing the underlying metric does not help and any
universal protocol has PoA of , which is tight. We attack this
problem by developing new techniques that employ powerful tools from extremal
combinatorics, and more specifically Ramsey Theory in high dimensional
hypercubes.
Then we switch to the stochastic model, where each player is independently
activated. We show that there exists a randomized ordered protocol that
achieves constant PoA. By using standard derandomization techniques, we produce
a deterministic ordered protocol with constant PoA.Comment: This version has additional results about stochastic inpu
Adaptive Approximation Algorithms for Ranking, Routing and Classification
This dissertation aims to consider different problems in the area of stochastic optimization, where we are provided with more information about the instantiation of the stochastic parameters over time. With uncertainty being an inseparable part of every industry, several applications can be modeled as discussed. In this dissertation we focus on three main areas of applications: 1) ranking problems, which can be helpful for modeling product ranking, designing recommender systems, etc., 2) routing problems which can cover applications in delivery, transportation and networking, and 3) classification problems with possible applications in medical diagnosis and chemical identification. We consider three types of solutions for these problems based on how we want to deal with the observed information: static, adaptive and a priori solutions. In Chapter II, we study two general stochastic submodular optimization problems that we call Adaptive Submodular Ranking and Adaptive Submodular Routing. In the ranking version, we want to provide an adaptive sequence of weighted elements to cover a random submodular function with minimum expected cost. In the routing version, we want to provide an adaptive path of vertices to cover a random scenario with minimum expected length. We provide (poly)logarithmic approximation algorithms for these problems that (nearly) match or improve the best-known results for various special cases. We also implemented different variations of the ranking algorithm and observed that it outperforms other practical algorithms on real-world and synthetic data sets. In Chapter III, we consider the Optimal Decision Tree problem: an identification task that is widely used in active learning. We study this problem in presence of noise, where we want to perform a sequence of tests with possible noisy outcomes to identify a random hypothesis. We give different static (non-adaptive) and adaptive algorithms for this task with almost logarithmic approximation ratios. We also implemented our algorithms on real-world and synthetic data sets and compared our results with an information theoretic lower bound to show that in practice, our algorithms' value is very close to this lower bound. In Chapter IV, we focus on a stochastic vehicle routing problem called a priori traveling repairman, where we are given a metric and probabilities of each vertices being active. We want to provide an a priori master tour originating from the root that is shortcut later over the observed active vertices. Our objective is to minimize the expected total wait time of active vertices, where the wait time of a vertex is defined as the length of the path from the root to this vertex. We consider two benchmarks to evaluate the performance of an algorithm for this problem: optimal a priori solution and the re-optimization solution. We provide two algorithms to compare with each of these benchmarks that have constant and logarithmic approximation ratios respectively.PHDIndustrial & Operations EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155058/1/navidi_1.pd