Speeding-up Dynamic Programming with Representative Sets - An Experimental Evaluation of Algorithms for Steiner Tree on Tree Decompositions

Dynamic programming on tree decompositions is a frequently used approach to solve otherwise intractable problems on instances of small treewidth. In recent work by Bodlaender et al., it was shown that for many connectivity problems, there exist algorithms that use time, linear in the number of vertices, and single exponential in the width of the tree decomposition that is used. The central idea is that it suffices to compute representative sets, and these can be computed efficiently with help of Gaussian elimination. In this paper, we give an experimental evaluation of this technique for the Steiner Tree problem. A comparison of the classic dynamic programming algorithm and the improved dynamic programming algorithm that employs the table reduction shows that the new approach gives significant improvements on the running time of the algorithm and the size of the tables computed by the dynamic programming algorithm, and thus that the rank based approach from Bodlaender et al. does not only give significant theoretical improvements but also is a viable approach in a practical setting, and showcases the potential of exploiting the idea of representative sets for speeding up dynamic programming algorithms

On optimal control policy for Probabilistic Boolean Network: a state reduction approach

BACKGROUND: Probabilistic Boolean Network (PBN) is a popular model for studying genetic regulatory networks. An important and practical problem is to find the optimal control policy for a PBN so as to avoid the network from entering into undesirable states. A number of research works have been done by using dynamic programming-based (DP) method. However, due to the high computational complexity of PBNs, DP method is computationally inefficient for a large size network. Therefore it is natural to seek for approximation methods. RESULTS: Inspired by the state reduction strategies, we consider using dynamic programming in conjunction with state reduction approach to reduce the computational cost of the DP method. Numerical examples are given to demonstrate both the effectiveness and the efficiency of our proposed method. CONCLUSIONS: Finding the optimal control policy for PBNs is meaningful. The proposed problem has been shown to be ∑ p 2 - hard . By taking state reduction approach into consideration, the proposed method can speed up the computational time in applying dynamic programming-based algorithm. In particular, the proposed method is effective for larger size networks.published_or_final_versio

Dynamic programming algorithm for the vehicle routing problem with time windows and EC social legislation

In practice, apart from the problem of vehicle routing, schedulers also face the problem of nding feasible driver schedules complying with complex restrictions on drivers' driving and working hours. To address this complex interdependent problem of vehicle routing and break scheduling, we propose a dynamic programming approach for the vehicle routing problem with time windows including the EC social legislation on drivers' driving and working hours. Our algorithm includes all optional rules in these legislations, which are generally ignored in the literature. To include the legislation in the dynamic programming algorithm we propose a break scheduling method that does not increase the time-complexity of the algorithm. This is a remarkable eect that generally does not hold for local search methods, which have proved to be very successful in solving less restricted vehicle routing problems. Computational results show that our method finds solutions to benchmark instances with 18% less vehicles and 5% less travel distance than state of the art approaches. Furthermore, they show that including all optional rules of the legislation leads to an additional reduction of 4% in the number of vehicles and of 1.5%\ud regarding the travel distance. Therefore, the optional rules should be exploited in practice

An improving dynamic programming algorithm to solve the shortest path problem with time windows

International audienceAn efficient use of dynamic programming requires a substantial reduction of the number of labels. We propose in this paper an efficient way of reducing the number of labels saved and dominance computing time. Our approach is validated by experiments on shortest path problem with time windows instances

A HJB-POD approach for the control of nonlinear PDEs on a tree structure

The Dynamic Programming approach allows to compute a feedback control for nonlinear problems, but suffers from the curse of dimensionality. The computation of the control relies on the resolution of a nonlinear PDE, the Hamilton-Jacobi-Bellman equation, with the same dimension of the original problem. Recently, a new numerical method to compute the value function on a tree structure has been introduced. The method allows to work without a structured grid and avoids any interpolation. Here, we aim to test the algorithm for nonlinear two dimensional PDEs. We apply model order reduction to decrease the computational complexity since the tree structure algorithm requires to solve many PDEs. Furthermore, we prove an error estimate which guarantees the convergence of the proposed method. Finally, we show efficiency of the method through numerical tests

Risk and optimal policies in bandit experiments

This paper provides a decision theoretic analysis of bandit experiments. The bandit setting corresponds to a dynamic programming problem, but solving this directly is typically infeasible. Working within the framework of diffusion asymptotics, we define a suitable notion of asymptotic Bayes risk for bandit settings. For normally distributed rewards, the minimal Bayes risk can be characterized as the solution to a nonlinear second-order partial differential equation (PDE). Using a limit of experiments approach, we show that this PDE characterization also holds asymptotically under both parametric and non-parametric distribution of the rewards. The approach further describes the state variables it is asymptotically sufficient to restrict attention to, and therefore suggests a practical strategy for dimension reduction. The upshot is that we can approximate the dynamic programming problem defining the bandit setting with a PDE which can be efficiently solved using sparse matrix routines. We derive near-optimal policies from the numerical solutions to these equations. The proposed policies substantially dominate existing methods such Thompson sampling. The framework also allows for substantial generalizations to the bandit problem such as time discounting and pure exploration motives