An analogy with the way ant colonies function has suggested the definition of a new computational paradigm, which we call Ant System. We propose it as a viable new approach to stochastic combinatorial optimization. The main characteristics of this model are positive feedback, distributed computation, and the use of a constructive greedy heuristic. Positive feedback accounts for rapid discovery of good solutions, distributed computation avoids premature convergence, and the greedy heuristic helps find acceptable solutions in the early stages of the search process. We apply the proposed methodology to the classical Traveling Salesman Problem (TSP), and report simulation results. We also discuss parameter selection and the early setups of the model, and compare it with tabu search and simulated annealing using TSP. To demonstrate the robustness of the approach, we show how the Ant System (AS) can be applied to other optimization problems like the asymmetric traveling salesman, the quadrat...

This paper introduces the ant colony system (ACS), a distributed algorithm that is applied to the traveling salesman problem (TSP). In the ACS, a set of cooperating agents called ants cooperate to find good solutions to TSP’s. Ants cooperate using an indirect form of communication mediated by a pheromone they deposit on the edges of the TSP graph while building solutions. We study the ACS by running experiments to understand its operation. The results show that the ACS outperforms other nature-inspired algorithms such as simulated annealing and evolutionary computation, and we conclude comparing ACS-3-opt, a version of the ACS augmented with a local search procedure, to some of the best performing algorithms for symmetric and asymmetric TSP’s.

Mobile Ad Hoc Networks (MANETs) provide rapidly deployable and self-configuring network capacity required in many critical applications, e.g., battlefields, disaster relief and wide area sensing. In this paper we study the problem of efficient data delivery in sparse MANETs where network partitions can last for a significant period. Previous approaches rely on the use of either long range communication which leads to rapid draining of nodes ’ limited batteries, or existing node mobility which results in low data delivery rates and large delays. In this paper, we describe a Message Ferrying (MF) approach to address the problem. MF is a mobility-assisted approach which utilizes a set of special mobile nodes called message ferries (or ferries for short) to provide communication service for nodes in the deployment area. The main idea behind the MF approach is to introduce non-randomness in the movement of nodes and exploit such non-randomness to help deliver data. We study two variations of MF, depending on whether ferries or nodes initiate proactive movement. The MF design exploits mobility to improve data delivery performance and reduce energy consumption in nodes. We evaluate the performance of MF via extensive ns simulations which confirm the MF approach is efficient in both data delivery and energy consumption under a variety of network conditions.

Abstract. We present a polynomial time approximation scheme for Euclidean TSP in fixed dimensions. For every fixed c � 1 and given any n nodes in � 2, a randomized version of the scheme finds a (1 � 1/c)-approximation to the optimum traveling salesman tour in O(n(log n) O(c) ) time. When the nodes are in � d, the running time increases to O(n(log n) (O(�dc))d�1). For every fixed c, d the running time is n � poly(log n), that is nearly linear in n. The algorithm can be derandomized, but this increases the running time by a factor O(n d). The previous best approximation algorithm for the problem (due to Christofides) achieves a 3/2-approximation in polynomial time. We also give similar approximation schemes for some other NP-hard Euclidean problems: Minimum Steiner Tree, k-TSP, and k-MST. (The running times of the algorithm for k-TSP and k-MST involve an additional multiplicative factor k.) The previous best approximation algorithms for all these problems achieved a constant-factor approximation. We also give efficient approximation schemes for Euclidean Min-Cost Matching, a problem that can be solved exactly in polynomial time. All our algorithms also work, with almost no modification, when distance is measured using any geometric norm (such as �p for p � 1 or other Minkowski norms). They also have simple parallel (i.e., NC) implementations.

Introduction A natural and well-studied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal

This paper describes a racing procedure for finding, in a limited amount of time, a configuration of a metaheuristic that performs as good as possible on a given instance class of a combinatorial optimization problem. Taking inspiration from methods proposed in the machine learning literature for model selection through cross-validation, we propose a procedure that empirically evaluates a set of candidate configurations by discarding bad ones as soon as statistically sufficient evidence is gathered against them. We empirically evaluate our procedure using as an example the configuration of an ant colony optimization algorithm applied to the traveling salesman problem.

This report describes an implementation of the Lin-Kernighan heuristic, one of the most successful methods for generating optimal or nearoptimal solutions for the symmetric traveling salesman problem. Computational tests show that the implementation is highly effective. It has found optimal solutions for all solved problem instances we have been able to obtain, including a 7397-city problem (the largest nontrivial problem instance solved to optimality today). Furthermore, the algorithm has improved the best known solutions for a series of large-scale problems with unknown optima, among these an 85900-city problem. 1.

Iterated Local Search has many of the desirable features of a metaheuristic: it is simple, easy to implement, robust, and highly effective. The essential idea of Iterated Local Search lies in focusing the search not on the full space of solutions but on a smaller subspace defined by the solutions that are locally optimal for a given optimization engine. The success of Iterated Local Search lies in the biased sampling of this set of local optima. How effective this approach turns out to be depends mainly on the choice of the local search, the perturbations, and the acceptance criterion. So far, in spite of its conceptual simplicity, it has lead to a number of state-of-the-art results without the use of too much problem-specific knowledge. But with further work so that the different modules are well adapted to the problem at hand, Iterated Local Search can often become a competitive or even state of the art algorithm. The purpose of this review is both to give a detailed description of this metaheuristic and to show where it stands in terms of performance. O.M. acknowledges support from the Institut Universitaire de France. This work was partially supported by the “Metaheuristics Network”, a Research Training Network funded by the Improving Human Potential programme of the CEC, grant HPRN-CT-1999-00106. The information provided is the sole responsibility of the authors and does not reflect the Community’s opinion. The Community is not responsible for any use that might be made of data appearing in this publication. 1 1

combine both communication and mobility capabilities. With mobility in devices, we envision a new class of proactive networks that are able to adapt themselves, via physical movement, to meet the needs of applications. To fully realize these opportunities, effective control of device mobility and the interaction between devices is needed. In this paper, we consider the Message Ferrying (MF) scheme which exploits controlled mobility to transport data in delay-tolerant networks, where end-to-end paths may not exist between nodes. In the MF scheme, a set of special mobile nodes called message ferries are responsible for carrying data for nodes in the network. We study the use of multiple ferries in such networks, which may be necessary to address performance and robustness concerns. We focus on the design of ferry routes. With the possibilities of interaction between ferries, the route design problem is challenging. We present algorithms to calculate routes such that the traffic demand is met and the data delivery delay is minimized. We evaluate these algorithms under a variety of network conditions via simulations. Our goal is to guide the design of MF systems and understand the tradeoff between the incurred cost of multiple ferries and the improved performance. We show that the performance scales well with the number of ferries in terms of throughput, delay and resource requirements in both ferries and nodes. Index Terms — System design, Simulations

INTRODUCTION 8.1 This chapter surveys approximation algorithms for hard geometric problems. The problems we consider typically take inputs that are point sets or polytopes in two- or three-dimensional space, and seek optimal constructions, (which may be trees, paths, or polytopes). We limit attention to problems for which no polynomial-time exact algorithms are known, and concentrate on bounds for worst-case approximation ratios, especially bounds that depend intrinsically on geometry. We illustrate our intentions with two well-known problems. Given a finite set of points S in the plane, the Euclidean traveling salesman problem asks for the shortest tour of S. Christofides' algorithm achieves approximation ratio 3 2 for this problem, meaning that it always computes a tour of length at most three-halves the length of the optimal tour. This bound depends only on the triangle inequality, so Christofides' algorit