Approximation Algorithms for Route Planning with Nonlinear Objectives

We consider optimal route planning when the objective function is a general nonlinear and non-monotonic function. Such an objective models user behavior more accurately, for example, when a user is risk-averse, or the utility function needs to capture a penalty for early arrival. It is known that as nonlinearity arises, the problem becomes NP-hard and little is known about computing optimal solutions when in addition there is no monotonicity guarantee. We show that an approximately optimal non-simple path can be efficiently computed under some natural constraints. In particular, we provide a fully polynomial approximation scheme under hop constraints. Our approximation algorithm can extend to run in pseudo-polynomial time under a more general linear constraint that sometimes is useful. As a by-product, we show that our algorithm can be applied to the problem of finding a path that is most likely to be on time for a given deadline.Comment: 9 pages, 2 figures, main part of this paper is to be appear in AAAI'1

On rr-Simple kk-Path

An $r$-simple $k$-path is a {path} in the graph of length $k$ that passes through each vertex at most $r$ times. The $r$-SIMPLE $k$-PATH problem, given a graph $G$ as input, asks whether there exists an $r$-simple $k$-path in $G$. We first show that this problem is NP-Complete. We then show that there is a graph $G$ that contains an $r$-simple $k$-path and no simple path of length greater than $4\log k/\log r$. So this, in a sense, motivates this problem especially when one's goal is to find a short path that visits many vertices in the graph while bounding the number of visits at each vertex. We then give a randomized algorithm that runs in time $$\mathrm{poly}(n)\cdot 2^{O( k\cdot \log r/r)}$$ that solves the $r$-SIMPLE $k$-PATH on a graph with $n$ vertices with one-sided error. We also show that a randomized algorithm with running time $\mathrm{poly}(n)\cdot 2^{(c/2)k/ r}$ with $c<1$ gives a randomized algorithm with running time \poly(n)\cdot 2^{cn} for the Hamiltonian path problem in a directed graph - an outstanding open problem. So in a sense our algorithm is optimal up to an $O(\log r)$ factor

The Lazy Bureaucrat Scheduling Problem

We introduce a new class of scheduling problems in which the optimization is performed by the worker (single ``machine'') who performs the tasks. A typical worker's objective is to minimize the amount of work he does (he is ``lazy''), or more generally, to schedule as inefficiently (in some sense) as possible. The worker is subject to the constraint that he must be busy when there is work that he can do; we make this notion precise both in the preemptive and nonpreemptive settings. The resulting class of ``perverse'' scheduling problems, which we denote ``Lazy Bureaucrat Problems,'' gives rise to a rich set of new questions that explore the distinction between maximization and minimization in computing optimal schedules.Comment: 19 pages, 2 figures, Latex. To appear, Information and Computatio

On the Placement of Wi-Fi Access Points for Indoor Localization

Nowadays, the more and more popular location based applications require accurate position information even in indoor environments. Wireless technologies can be used to derive positioning data. Especially, the Wi-Fi technology is popular for indoor localization because the existing and almost ubiquitous worldwide Wi-Fi infrastructure can be reused lowering the expenses. However, the primary purpose of these Wi-Fi systems is different from being used for positioning services, thus the accuracy they provide might be low. This accuracy can be increased by carefully placing the Wi-Fi access points to cover the given territory appropriately. In this paper, we propose a simulated annealing based method to find, in a given area, the optimal number and placement of Wi-Fi access points to be used for indoor positioning. We investigate the performance of our method via simulations

Optimization of Wi-Fi Access Point Placement for Indoor Localization

The popularity of location based applications is undiminished today. They require accurate location information which is a challenging issue in indoor environments. Wireless technologies can help derive indoor positioning data. Especially, the Wi-Fi technology is a promising candidate due to the existing and almost ubiquitous Wi-Fi infrastructure. The already deployed Wi-Fi devices can also serve as reference points for localization eliminating the cost of setting up a dedicated system. However, the primary purpose of these Wi-Fi systems is data communication and not providing location services. Thus their positioning accuracy might be insufficient. This accuracy can be increased by carefully placing the Wi-Fi access points to cover the given territory properly. In this paper, our contribution is a method based on simulated annealing, what we propose to find the optimal number and placement of Wi-Fi access points with regard to indoor positioning. We investigate its performance in a real environment scenario via simulations