Optimal paths in graphs with stochastic or multidimensional weights (original) (raw)
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Shortest paths in stochastic networks
Proceedings. 2004 12th IEEE International Conference on Networks (ICON 2004) (IEEE Cat. No.04EX955), 2004
This paper discusses the sensitivity of network flows to uncertain link state information for various routing protocols. We show that the choice of probability distribution for the link metrics for a given network can have markedly different effects on the probabilities of path selection. Exact results are obtained for these probabilities but their computation is NP-hard. We provide simulation results for three networks to illustrate the sensitivity of shortest paths to different link metric distributions. We provide results for mean path costs and the k-shortest path algorithm as a comparison.
An analysis of stochastic shortest path problems
Mathematics of Operations Research, 1991
We consider a stochastic version of the classical shortest path problem whereby for each node of a graph, we must choose a probability distribution over the set of successor nodes so as to reach a certain destination node with minimum expected cost. The costs of transition between successive nodes can be positive as well as negative. We prove natural generalizations of the standard results for the deterministic shortest path problem, and we extend the corresponding theory for undiscounted finite state Markovian decision problems by removing the usual restriction that costs are either all nonnegative or all nonpositive.
Optimal Paths in Probabilistic Networks
Journal of Mathematical Sciences, 2004
The random graph theory was initially proposed by P. Erdős and A. Rényi in the 1950s-1960s. More recently, B. Bollobás presented the first systematic and extensive treatment of results in the theory of random graphs. Associating to each edge of a random graph a real random variable, we obtain a probabilistic network. The determination of an optimal path between two nodes in a probabilistic network was first studied by H. Frank in 1969. Since then few theoretical results have become known, even though there is a recognizable applicability of this type of network to real problems, namely, to social and telecommunication networks. The mathematical model proposed in this paper maximizes the expected value of a utility function over a directed random network, where the costs related to the arcs are real random variables following Gaussian distributions. We consider the linear, quadratic, and exponential cases, presenting a theoretical formulation based on multi-criteria models as well as the resulting algorithms and computational tests.
A note on the stochastic shortest-route problem
Naval Research Logistics Quarterly
This paper develops an algorithm for a "shortest route" network problem in which it is desired to find the path which yields the shortest expected distance through the network. It is assumed that if a particular arc is chosen, then there is a finite probability that an adjacent arc will be traversed instead. Backward induction is used and appropriate recursion formulae are developed. A numerical example is provided.
Shortest paths in networks with correlated link weights
2015 IEEE Conference on Computer Communications Workshops (INFOCOM WKSHPS), 2015
Solving the shortest path problem is important in achieving high performance or to efficiently utilize resources in various kinds of networks, e.g., data communication networks and transportation networks. Fortunately, under independent additive link weights, this problem is solvable in polynomial time. However, in many real-life networks, the link weights (e.g., delay, bandwidth, failure probability) are often correlated due to spatial or temporal dependencies. These correlated link weights together might behave in a different manner and are not always additive. In this paper, we first propose two correlated link-weight models, namely (i) the deterministic correlated model and (ii) the (log-concave) stochastic correlated model. Subsequently, we study the shortest path problem under these two correlated models. We prove that the shortest path problem is NP-hard under the deterministic correlated model, and even cannot be approximated to arbitrary degree in polynomial time. On the other hand, we show that the shortest path problem is polynomial-time solvable under a nodal deterministic correlated model. Finally, we show that the shortest path problem under the (log-concave) stochastic correlated model can be solved by convex optimization.
Distributionally robust stochastic shortest path problem
Electronic Notes in Discrete Mathematics, 2013
This paper considers a stochastic version of the shortest path problem, the Distributionally Robust Stochastic Shortest Path Problem(DRSSPP) on directed graphs. In this model, the arc costs are deterministic, while each arc has a random delay. The mean vector and the second-moment matrix of the uncertain data are assumed known, but the exact information of the distribution is unknown. A penalty occurs when the given delay constraint is not satisfied. The objective is to minimize the sum of the path cost and the expected path delay penalty. As it is NP-hard, we approximate the DRSSPP with a semidefinite programming (SDP for short) problem, which is solvable in polynomial time and provides tight lower bounds.
Shortest paths in stochastic networks with correlated link costs
Computers & Mathematics with Applications, 2005
The objective is to minimize expected travel time from any origin to a specific destination in a congestible network with correlated link costs. Each link is assumed to be in one of two possible conditions. Conditional probability density functions for link travel times are assumed known for each condition. Conditions over the traversed links are taken into account for determining the optimal routing strategy for the remaining trip. This problem is treated as a multi-stage adaptive feedback control process. Each stage is described by the physical state (the location of the current decision point) and the information state (the service level of the previously traversed links). Proof of existence and uniqueness of the solution to the basic dynamic programming equations and a solution procedure are provided.
The shortest path problem in the stochastic networks with unstable topology
SpringerPlus, 2016
Background The deterministic shortest path problem has been studied extensively and applied in many fields of optimization; there are polynomial time algorithms to solve the deterministic shortest path problem (Dijkstra 1959; Bellman 1958; Orlin et al. 2010). However, paths in networks should be reliable to transmit flow from a source node to a destination node especially in delay sensitive networks. The best connection helps to avoid traffic congestion in networks. So, the arrival probability is used to evaluate the reliability of paths and it has been considered as an optimality index of the stochastic shortest path length (Bertsekas and Tsitsiklis 1991; Fan et al. 2005; Kulkarni 1986; Shirdel and Abdolhosseinzadeh 2016). The stochastic shortest path problem (SSP) is defined as the best path with stochastic optimality condition. Liu (2010) assumed the arc lengths to be uncertain variables. Pattanamekar et al. (2003) considered the individual travel time variance and the mean travel time forecasting error. Also, Hutson and Shier (2009) and Rasteiro and Anjo (2004) supposed two criteria: mean and variance of path length. Fan et al. (2005) assumed known conditional probabilities for link travel times that each link
Transportation Research Record: Journal of the Transportation Research Board, 2014
The aim of this study was to solve the minimum path travel time budget (MPTTB) problem, in which the travel time budget was the reliability index. This index was defined as the sum of the mean path travel time and the scaled standard deviation, which included the covariance matrix to consider correlation. Two existing solution methods in the literature, the outer approximation algorithm and Monte Carlo simulation method, were applied to solve the MPTTB problem. The former method approximated the hard nonlinear constraint of the MPTTB problem by a series of linear cuts generated iteratively and repeatedly solved a mixed integer program. The latter method, which was a simulation-based method, included two stages. The first stage founded a set of candidate paths, and the second stage generated the distribution of travel times for the existing paths in the candidate set. The numerical results for these two solution methods were conducted on the Chicago sketch network, and results showed that the methods found comparable solutions though they have respective advantages and drawbacks. Although the outer approximation algorithm demonstrated promising performance, it still relied on repeatedly solving a mixed integer program (subproblem) with a commercial solver, which could be a challenging task in its own right.