Graph Planning with Expected Finite Horizon

Graph planning with expected finite horizon

Some dynamical properties of sequentially acquired information

In this paper we obtain two closely related theorems that essentially say that no matter what information metric is used, on the average the value of the accumulated information at stopping time is bounded by a multiple of the expected stopping time. These results are also independent of the particular stopping strategy employed although they do require that the expected stopping time be finite. These results, along with a general type of stopping strategy based on incremental information, are given. Later we apply our general theorem to a specific stopping strategy associated with the GIS model. Although we concentrate on the problem of stopping, the information function on which this stopping decision is based can also be used to choose the COA for the next cycle of the feedback loop. We apply our results to an estimation problem involving the well known Shannon-Weiner measure of information. Since our theorems require that the expected stopping times be finite, sometime is devoted to a discussion of necessary and sufficient conditions for finite expected stopping times.

Markov chains are the de facto finite-state model for stochastic dynamical systems, and Markov decision processes (MDPs) extend Markov chains by incorporating non-deterministic behaviors. Given an MDP and rewards on states, a classical optimization criterion is the maximal expected total reward where the MDP stops after T steps, which can be computed by a simple dynamic programming algorithm. We consider a natural generalization of the problem where the stopping times can be chosen according to a probability distribution, such that the expected stopping time is T, to optimize the expected total reward. Quite surprisingly we establish inter-reducibility of the expected stopping-time problem for Markov chains with the Positivity problem (which is related to the well-known Skolem problem), for which establishing either decidability or undecidability would be a major breakthrough. Given the hardness of the exact problem, we consider the approximate version of the problem: we show that it can be solved in exponential time for Markov chains and in exponential space for MDPs.

Markov chains are the de facto finite-state model for stochastic dynamical systems, and Markov decision processes (MDPs) extend Markov chains by incorporating non-deterministic behaviors. Given an MDP and rewards on states, a classical optimization criterion is the maximal expected total reward where the MDP stops after T steps, which can be computed by a simple dynamic programming algorithm. We consider a natural generalization of the problem where the stopping times can be chosen according to a probability distribution, such that the expected stopping time is T, to optimize the expected total reward. Quite surprisingly we establish inter-reducibility of the expected stopping-time problem for Markov chains with the Positivity problem (which is related to the well-known Skolem problem), for which establishing either decidability or undecidability would be a major breakthrough. Given the hardness of the exact problem, we consider the approximate version of the problem: we show that it can be solved in exponential time for Markov chains and in exponential space for MDPs.

For planning to come of age, plans must be judged by a measure of quality, such as the total cost of actions. This paper describes an optimal-cost planner which guarantees global optimality whenever the planning problem has a solution. We code the extraction of an optimal plan, from a planning graph with a fixed number k of levels, as a weighted constraint satisfaction problem (WCSP). The specific structure of the resulting WCSP means that a state-of-the-art exhaustive solver was able to find an optimal plan in planning graphs containing several thousand nodes. Thorough experimental investigations demonstrated that using the planning graph in optimal planning is a practical possibility for problems of moderate size, although not competitive, in terms of computation time, with optimal state-space-search planners. Our general conclusion is, therefore, that planning-graph-based optimal planning is not the most efficient method for cost-optimal planning. Nonetheless, the notions of indispensable (sets of) actions and too-costly actions introduced in this paper have various potential applications in optimal planning. These actions can be detected very rapidly by analysis of the relaxed planning graph.

In this paper, we obtain a recursive formula for the density of the two‐sided Parisian stopping time. This formula does not require any numerical inversion of Laplace transforms, and is similar to the formula obtained for the one‐sided Parisian stopping time derived in Dassios and Lim. However, when we study the tails of the two distributions, we find that the two‐sided stopping time has an exponential tail, while the one‐sided stopping time has a heavier tail. We derive an asymptotic result for the tail of the two‐sided stopping time distribution and propose an alternative method of approximating the price of the two‐sided Parisian option.

Summary Generalized sequential probability-ratio procedures are defined for dependent and non-identically distributed random variables. For these procedures, conditions are found implying that the stopping time is finite with probability 1 and the expected stopping time is finite. These results are applied to rank order problems.

The Travelling Salesman Problem (TSP) is a classical NP-hard optimisation problem. There exist, however, special cases of the TSP that can be solved in polynomial time. Many of the well-known TSP special cases have been characterized by imposing special four-point conditions on the underlying distance matrix. Probably the most famous of these special cases is the TSP on a Monge matrix, which is known to be polynomially solvable (as are some other generally NP-hard problems restricted to this class of matrices). By relaxing the four-point conditions corresponding to Monge matrices in different ways, one can define other interesting special cases of the TSP, some of which turn out to be polynomially solvable, and some NP-hard. However, the complexity status of one such relaxation, which we call one-sided Monge TSP (also known as the TSP on a relaxed Supnick matrix), has remained unresolved. In this note, we show that this version of the TSP problem is NP-hard. This completes the full classification of all possible four-point conditions for symmetric TSP.

The paper is devoted to the construction of mathematical models for problems on the shortest cycles and paths, that pass through a given number of nodes of a directed graph. Such cycles and paths are called k-node, where 1<k <n, n is the number of nodes in the graph. Section 1 formulates two problems for finding the shortest k-node cycle – a mixed Boolean and linear programming problem and a discrete programming problem. Both problems include constraints from the classical assignment problem, describing a one-time entry into a node and a one-time exit from a node for those nodes through which the cycle passes. The cycle connectivity in the first problem is ensured by modeling the flow problem, and in the second problem, it is ensured by using the A. Tucker constraints for the travelling salesman problem. Section 2 establishes a connection between the formulations of both problems from Section 1 and the travelling salesman problem and investigates the efficiency of their solution using modern versions of gurobi and cplex programs and the AMPL modeling language. Section 3 contains the formulation of the shortest k-node path problem, which is represented by a mixed Boolean and linear programming problem. With its help, the optimal routes were found for visiting the wine-making points of the Malopolskie Wine Route in the direction Lviv-Wroclaw-Lviv (Section 4). Here a map for the 20 most visited wine-making points of the Malopolskie Wine Route and a table of the distances between them and the distances from them to Lviv and Wroclaw, calculated using the Google Maps web service, are presented. The developed mathematical models of the problems of finding the shortest k-node paths and cycles and the developed software in the AMPL modeling language can be used for the design and arrangement of technical objects, optimization of the transportation of products, analysis and forecasting of economic processes, determination of optimal routes when planning passenger and freight traffic, optimal organization of the process of managing a set of transactions and queries during their implementation in network databases and other classes of applied optimization problems. Keywords: digraph, shortest path, Boolean variable, linear programming, Hamiltonian cycle, Hamiltonian path, travelling salesman problem, AMPL, gurobi, cplex.

Traveling salesman problem under categorization

Logistics means management of goods transportation from one place into another in an effective way. Most of the logistic companies is very effective in transportation of goods through proper design of logistics network. When the logistics network design is effective, then the delivery of goods to the right place at right time is effective. In general travelling salesman problem (TSP) supports to design logistics network and plays a vital role in effective logistics management. Hence, solving TSP in polynomial time will greatly influences the proper design logistics network. As a result the delivering of goods will be very effective. Thus the objective of this project is to solve TSP using optimization algorithm in order to design an effective logistics network. Many researchers have proposed different kinds of optimization algorithms to solve NP hard problem TSP. But still there is a need to solve TSP in polynomial time and also to handle uncertainty situation. Thus the aim of this project is to use adaptive based Donkey-Smugglers optimization (DSO) algorithm to solve TSP problem in polynomial time and handle uncertainty events in more effective. The salient feature of this proposed system is able to handle uncertainty situation with the help of adding adaptive part in DSO algorithm to reduce time delay in delivery of goods. Since logistics network is heavily depends upon time, traffic, load weight, competency of driver etc. Hence there is a possible for uncertainty happens during delivering of goods. In order to handle this uncertainty situation, the project aims to solve TSP problem using adaptive based DSO algorithm. Meanwhile the proposed system also compares the results with other adaptive optimization algorithms namely Genetic Algorithm (GA) and Ant Colony Optimization (ACO). The results have been proved that the Adaptive based DSO algorithm works well even in uncertainty situation and able to solve TSP in polynomial time. As well as the proposed system achieves minimum average time taken even when there is large number of instances. Hence the proposed system reduces time delay effectively and helps to deliver goods at right place at right minimum time.

This dissertation studies the integrality gap of linear programming relaxations of integer programs. The integrality gap of a continuous relaxation of the sets of latticepoints corresponding to integer feasible solutions is the worst case ratio between the cost of an integer feasible solution and the optimal value of the continuous relaxation.The main focus in the ?first part of the thesis is on the Traveling Salesperson Problem (TSP) and the 2-edge-connected multigraph problem (2ECM). In TSP and 2ECM we are given n vertices with costs on pairs of vertices. We consider cost functions obeying triangle inequality. In TSP the goal is to fi?nd the minimum cost Hamiltonian cycle andin the 2ECM the goal is to ?find the minimum cost 2-edge-connected subgraph. Both problems can be formulated via a linear programming relaxation known as the subtour elimination relaxation. The most general case for TSP and 2ECM has resisted approximation algorithms (and upper bounds on the integrality gap with the subtourelimination relaxation) better than 3/2 for decades.In Chapter 3 we consider TSP and 2ECM on node-weighted graphs. These are instances where the cost on the pairs of vertices arise from a shortest path between the pair in a node-weighted graph, a graph with edge weights arising from adding the costs of its endpoints. First we show that for 3-edge-connected cubic graphs, there is a 7/5 approximation algorithm for the node-weighted TSP and a 13/10-approximation for the node-weighted 2ECM. The main tool for both algorithms is the fact that 3-edgeconnectedcubic graphs contain 2-factors covering all their small edge cuts. We extend this result to subcubic graphs by providing a decomposition of a point of the subtour elimination relaxation into a convex combination of connected multigraphs, each covering 2-edge cuts an even number of times. An application of this decomposition leads to a17/12 -approximation algorithm for node-weighted 2ECM on subcubic graphs. Chapter 4 focuses on the Uniform Cover Problem for TSP and 2ECM. We establish this framework as a way to approach the most general case of TSP and 2ECM. As a ?first result, we give the ?first positive answer to Seb}o et al. [SBS14] regarding the uniform cover problem for TSP by showing that for a 3-edge-connected cubic graph, the incidence vector of G multiplied by 18=19 can be decomposed into a convex combination of solutions for the TSP: this is equivalent to a 27/19-approximation for TSP on such instances. We also provide a 45/34 -approximation for 2ECM on such instances. This is the first bound below 43 that can be proved via an efficient rounding algorithm. Improving this factor further requires a technique commonly known as \gluing". We show how gluing on 3-edge cuts reduces our problems to more structured instances. For such structured instances we use a novel application of a rainbow 1-tree decomposition that serves a top-down coloring algorithm in order to improve the factor of 45/34 <b>≈ </b>1:323 to 123/94 <b>≈ </b>1:308. In Chapter 5 our focus is on half-integer points of the subtour elimination relaxationmotivated by the conjecture of Schalekamp, Williamson, van Zuylen [SWvZ13] that the largest integrality gap is achieved for instances where the optimal solution of the subtourelimination relaxation is half-integer. Our focus is on fundamental classes that are a class of interesting yet highly structured points in the subtour elimination relaxation.In particular, we study half-square points and half-triangle points. For half-squarepoints we provide a 9/7 -approximation for 2ECM and for half-triangle points we show a ( 6/5 + 1/120 )-approximation for 2ECM. In Chapter 6 we investigate the possibility of gluing the solutions for T SP over 3-edge cuts. Gluing over 3-edge cuts has proven to be successful for 2-edge-connected subgraphs but there is not much known in this direction for gluing connected multigraphs. We introduce a novel approach of gluing solutions to the TSP based on different parts ofa tour: (i) the connected skeleton of a solution which is a connected subgraph and (ii) the parity correction part of the solution that augments the connected skeleton into anEulerian connected multigraph. Using this approach we show that for a half-integer point x of the subtour elimination relaxation, we can reduce the usage of edges with x-value 1 from the 3/2 of Christo?des' algorithm to 3/2 - 1/20 while keeping the usage of edges with x-value of 1 2 the same as Christo?des' algorithm. A direct consequence of this result is for the Uniform Cover Problem for TSP, where we show that for a 3-edge-connected cubic graph, the incidence vector of G multiplied by 17/18 can be decomposed into a convex combination of solutions for the TSP: In this way we improvethe 27/19 -approximation algorithm in Chapter 4 to a 17/12 -approximation algorithm for TSP on these instances.In the fi?nal chapter of this thesis, we focus on general binary integer programs (binary IPs) and show an efficient algorithm, called the Fractional Decomposition Tree Algorithm (FDT), that provides an upper bound on the integrality gap of an instance of a binary IP with its linear programming relaxation. As a stepping stone, we design an efficient algorithm for ?finding a feasible integer solution to binary IPs with bounded integrality gap which may be of independent interest. We extend FDT to ?find convex combinations of 2-edge-connected multigraphs which is a non-binary problem. We run experiments and compare upper bounds provided by FDT with that of polyhedral version of Christo?des' algorithm.

On: Travelling salesman problem under categorization: Operations Research Letters 12 (1992) 89–95

The sequential testing of more than two hypotheses has important applications in direct-sequence spread spectrum signal acquisition, multiple-resolution-element radar, and other areas. A useful sequential test which we term the MSPRT is studied in this paper. The test is shown to be a generalization of the sequential probability ratio test. Under Bayesian assumptions, it is argued that the MSPRT approximates the much more complicated optimal test when error probabilities are small and expected stopping times are large. Bounds on error probabilities are derived, and asymptotic expressions for the stopping time and error probabilities are given. A design procedure is presented for determining the parameters of the MSPRT. Two examples involving Gaussian densities are included, and comparisons are made between simulation results and asymptotic expressions. Comparisons with Bayesian fixed sample size tests are also made, and it is found that the MSPRT requires two to three times fewer samples on average.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>