We consider a problem which has received considerable attention in systems literature because of its applications to routing in delay tolerant networks and replica placement in distributed storage systems. In abstract terms the problem can be stated as follows: Given a random variable @math generated by a known product distribution over @math and a target value @math , output a non-negative vector @math , with @math , which maximizes the probability of the event @math . This is a challenging non-convex optimization problem for which even computing the value @math of a proposed solution vector @math is #P-hard. We provide an additive EPTAS for this problem which, for constant-bounded product distributions, runs in @math time and outputs an @math -approximately optimal solution vector @math for this problem. Our approach is inspired by, and extends, recent structural results from the complexity-theoretic study of linear threshold functions. Furthermore, in spite of the objective function being non-smooth, we give a PTAS while previous work for such objective functions has typically led to a PTAS. We believe our techniques may be applicable to get unicriterion PTAS for other non-smooth objective functions.
Previous Work on the Problem. The stochastic design problem (P) stated above was formulated explicitly in the work of @cite_4 . That work was motivated by the problem of routing in Delay Tolerant Networks @cite_44 . These networks are characterized by a lack of consistent end-to-end paths, due to interruptions that may be either planned or unplanned, and selecting routing paths is considered to be one of the most challenging problems. The authors of @cite_4 reduce the route selection problem to Problem (P) in a range of settings of interest, and study the structure of the optimal partition as well as its computational complexity, albeit with inconclusive theoretical results.
@cite_44 @cite_4
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This paper studies the problem of finding a priori shortest paths to guarantee a given likelihood of arriving on-time in a stochastic network. Such "reliable" paths help travelers better plan their trips to prepare for the risk of running late in the face of stochastic travel times. Optimal solutions to the problem can be obtained from local-reliable paths, which are a set of non-dominated paths under first-order stochastic dominance. We show that Bellman's principle of optimality can be applied to construct local-reliable paths. Acyclicity of local-reliable paths is established and used for proving finite convergence of solution procedures. The connection between the a priori path problem and the corresponding adaptive routing problem is also revealed. A label-correcting algorithm is proposed and its complexity is analyzed. A pseudo-polynomial approximation is proposed based on extreme-dominance. An extension that allows travel time distribution functions to vary over time is also discussed. We show that the time-dependent problem is decomposable with respect to arrival times and therefore can be solved as easily as its static counterpart. Numerical results are provided using typical transportation networks. In this article, the following stochastic vehicle routing problem is considered. Each customer has a known probability of presence and a random demand. This problem arises in several contexts, e.g., in the design of less-than-truckload collection routes. Because of uncertainty, it may not be possible to follow vehicle routes as planned. Using a stochastic programming framework, the problem is solved in two stages. In a first stage, planned collection routes are designed. In a second stage, when the set of present customers is known, these routes are followed as planned by skipping the absent customers. Whenever the vehicle capacity is attained or exceeded, the vehicle returns to the depot and resumes its collections along the planned route. This generates a penalty. The problem is to design a first stage solution in order to minimize the expected total cost of the second state solution. This is formulated as a stochastic integer program, and solved for the first time to optimality by means of an integer L-shaped method. We consider the problem of routing in a delay tolerant network (DTN) in the presence of path failures. Previous work on DTN routing has focused on using precisely known network dynamics, which does not account for message losses due to link failures, buffer overruns, path selection errors, unscheduled delays, or other problems. We show how to split, replicate, and erasure code message fragments over multiple delivery paths to optimize the probability of successful message delivery. We provide a formulation of this problem and solve it for two cases: a 0 1 (Bernoulli) path delivery model where messages are either fully lost or delivered, and a Gaussian path delivery model where only a fraction of a message may be delivered. Ideas from the modern portfolio theory literature are borrowed to solve the underlying optimization problem. Our approach is directly relevant to solving similar problems that arise in replica placement in distributed file systems and virtual node placement in DHTs. In three different simulated DTN scenarios covering a wide range of applications, we show the effectiveness of our approach in handling failures. In this paper, we introduce a distributed dynamic routing algorithm in multi-hop cognitive radio (CR) networks, in which secondary users (SUs) want to minimize their interference to the primary users (PUs) while keeping the delay along the route low. We employ a cognitive pilot channel (CPC) for SUs to be able to access the information about PUs, including PUs' locations and channel conditions. Medial axis with a relaxation factor is used as a reference path for the routing, along which we develop a hierarchical structure for multiple sources to reach their destinations. We introduce a temporal and spatial dynamic non-cooperative game to model the interactions among the SUs as well as their influences on the PUs, and obtain by backward induction a set of mixed (behavioral) Nash equilibrium strategies. We also employ a multi-stage fictitious play learning algorithm for distributed routing, which minimizes the overall interference from the SUs to the PUs, as well as the average packet delay along the route from the SU nodes to their destinations. Simulation results show that our proposed algorithm can avoid congestion in the CR network and minimize delay while keeping the interference level low.