We are given a set of points p 1 ; . . . ; p n and a symmetric distance matrix (d ij ) giving the distance between p i and p j . We wish to construct a tour that minimizes P n i=1 `(i), where `(i) is the latency of p i , de ned to be the distance traveled before rst visiting p i . This problem is also known in the literature as the deliveryman problem or the traveling repairman problem. It arises in a number of applications including disk-head scheduling, and turns out to be surprisingly di erent from the traveling salesman problem in character. We give exact and approximate solutions to a number of cases, including a constant-factor approximation algorithm whenever the distance matrix satis es the triangle inequality.

In a general discrete-time market model with proportional transaction costs, we derive new expectation representations of the range of arbitrage-free prices of an arbitrary American option. The upper bound of this range is called the upper hedging price, and is the smallest initial wealth needed to construct a self-financing portfolio whose value dominates the option payoff at all times. A surprising feature of our upper hedging price representation is that it requires the use of randomized stopping times (Baxter and Chacon 1977), just as ordinary stopping times are needed in the absence of transaction costs. We also represent the upper hedging price as the optimum value of a variety of optimization problems. Additionally, we show a two-player game where at Nash equilibrium the value to both players is the upper hedging price, and one of the players must in general choose a mixture of stopping times. We derive similar representations for the lower hedging price as well.Our results make use of strong duality in linear programming.

We provide approximation algorithms for some capacitated vehicle routing and delivery problems. These problems can all be viewed as instances of the following k-Delivery TSP: given n source points and n sink points in a metric space, with exactly one item at each source, nd a minimum length tour by a vehicle of nite capacity k to pick up and deliver exactly one item to each sink. The only known approximation algorithm for this family of problems is the 2.5-approximation algorithm of Anily and Hassin 2] for the special case k = 1. For this case, we use matroid intersection to obtain a 2-approximation algorithm. Based on this algorithm and some additional lower bound arguments, we devise a 9.5-approximation for k-Delivery TSP with arbitrary nite k. We also present a 2-approximation algorithm for the case k = 1. We then initiate the study of dynamic variants of k-Delivery TSP that model problems in industrial robotics and other applications. Speci cally, we consider the situation where a robot arm (with nite or in nite capacity) must collect n point-objects moving in the Euclidean plane, and deliver them to the origin. The point-objects are moving in the plane with known, identical velocities { they might, for instance, be on a moving conveyor belt. We derive several useful structural properties that lead to constant-factor approximations for problems of this type that are relevant to the robotics application. Along the way, we show that Maximum Latency TSP is implicit in the dynamic problems, and that the natural \furthest neighbor" heuristic produces a good approximation for several notions of latency.

In the binomial tree model, we provide efficient algorithms for computing an accurate lower bound for the value of a European-style Asian option with either a fixed or a floating strike. These algorithms are inspired by the continuous-timeanalysis of Rogers and Shi [10]. Specifically we consider lower bounds on the option value that are given by the expectation of the conditional expectation of the payoff conditioned on some random variable Z. For a specific Z, Rogers and Shi estimate this conditional expectation numerically in continuous time, and show experimentally that their lower bound is very accurate. We consider a modified random variable Z that gives a strictly better lower bound. In addition, we show that this lower bound can be computed exactly in the n-step binomial tree model in time proportional to nW. We show that computing the approximation is equivalent to counting paths of various types, and that this can be done efficiently by a dynamic programming technique. We present other choices of Z that yield accurate and efficiently-computable lower bounds. We also show algorithms to compute a bound on the error of these approximations, so that we can compute an upper bound on the option value as well.

Multi-agent systems MAS are subject to performance bottlenecks in cases where agents cannot perform tasks by themselves due to insu cient resources. Solutions to such problems include passing tasks to others or agent migration to remote hosts. We propose agent cloning as a more comprehensive approach to the problem of local agent overloads. Agent cloning subsumes task transfer and agent mobility. According to our paradigm, agents may clone, pass tasks to others, die or merge. We discuss the requirements of implementing a cloning mechanism and its bene ts in a Multi-Agent System, and support our claims with simulation results.