New Complexity Results and Algorithms for the Minimum Tollbooth Problem

The inefficiency of the Wardrop equilibrium of nonatomic routing games can be eliminated by placing tolls on the edges of a network so that the socially optimal flow is induced as an equilibrium flow. A solution where the minimum number of edges are tolled may be preferable over others due to its ease of implementation in real networks. In this paper we consider the minimum tollbooth (MINTB) problem, which seeks social optimum inducing tolls with minimum support. We prove for single commodity networks with linear latencies that the problem is NP-hard to approximate within a factor of $1.1377$ through a reduction from the minimum vertex cover problem. Insights from network design motivate us to formulate a new variation of the problem where, in addition to placing tolls, it is allowed to remove unused edges by the social optimum. We prove that this new problem remains NP-hard even for single commodity networks with linear latencies, using a reduction from the partition problem. On the positive side, we give the first exact polynomial solution to the MINTB problem in an important class of graphs---series-parallel graphs. Our algorithm solves MINTB by first tabulating the candidate solutions for subgraphs of the series-parallel network and then combining them optimally

Noise Control And Speech Intelligibility Improvement Of A Toll Plaza

Vehicular toll roads are one component of many municipal transportation systems. Open communication windows, often used in tollbooths, facilitate essential communication and monetary transactions. However, the vehicle noise generated outside the booth is easily transmitted into the booth via the open window. Personnel working at toll collection plazas are exposed to extended, continuous traffic noise. Sustained noise levels of this nature may cause hearing loss, induce fatigue or stress, and reduce worker productivity. The annoyance and discomfort related to continuous noise exposure may create an unpleasant working condition and may affect the hospitality of the tollbooth operators and their attitude toward customers. Furthermore, the noise level may hinder communication with customers and may compromise safety. Reduction of the noise level and an improvement in speech intelligibility are highly desirable. The acoustics of a typical toll plaza and structural noise control strategies were modeled using a beam tracing technique. Noise control strategies involved the application sound absorbing material to the overhead canopy, the construction of sound absorbing partial barriers, and the treatment of tollbooth walls with sound absorbing material. In terms of noise control, the results suggest that the direct field is more important that the reflected field. The effects of active noise control (ANC) systems to reduce traffic noise and improve speech intelligibility at the toll plaza was investigated. The ANC systems included a range of headsets and a prototype external unit designed to create a local region of attenuation. Significant noise reduction can be achieved with a sealed, closed ear ANC headsets. However, the various systems seemed to have little positive effect upon speech intelligibility under traffic noise conditions. The result imply that the signal to noise ratio under toll plaza conditions is poor and that level overloading effects may further reduce intelligibility. Altered systems were modeled to improve the signal to noise ratio and reduce the noise level. The improved systems utilize a directional microphone and a sealed ANC headset. With a high order directional microphone, good speech intelligibility is achievable even in the presence of toll plaza vehicle noise

Microscopic Simulation On The Operation And Capacity Of Toll Plaza In Malaysia

Microscopic traffic simulation software has several applications, such as performance evaluation, plan improvements, traffic operation control, design, and transportation facility management. This study presents the application of the well-known traffic simulation software VISSIM in the operation of toll plazas in Malaysia. This study evaluates the overall toll operation of two types of closed system toll plazas in the Malaysian expressway to gain insight into the variables that influence toll operations, which in turn affect the actual capacity of toll plazas in terms of average and maximum queue length. VISSIM was used to build toll plaza models for the mainline and ramp toll plazas which are Juru and Jawi respectively, to study their toll operations and actual capacities. In order to simulate the toll operations at toll plazas, microscopic data were obtained for each vehicle arriving and departing the toll plazas through video recordings. Video recordings were taken from two sources. The first source was from the installed CCTV and the second source was from the PLUS CCTV cameras at the tollbooths. The collected field data of the Juru and Jawi toll plazas differed in terms of number of lanes, lane configuration, toll base fee, expressway location, traffic demand, and traffic composition characteristics. The toll plaza models were then calibrated according to the measure of effectiveness and key parameter to match real world toll operations at toll plazas. Results revealed that service time is the most important parameter for evaluating the toll operation of toll plazas. Moreover, service time for entry is much lower than the service time for exit. The findings indicated that the percentage of heavy vehicles in traffic flow has a significant impact on the queue lengths at the Juru and Jawi toll plazas. Apart from that, the models were used to predict the operation of toll plazas in the future upon implementation of full electronic toll collection (ETC). The results indicated that the implementation of full ETC at the entry of both the Juru and Jawi toll plazas did not improve the operations of the toll plazas. However, the implementation of full ETC at the exit significantly improved the toll operations. But, the implementation of full ETC at the exit of the Jawi toll plaza has negatively influenced the queue lengths of Touch 'n Go and Smart TAG lanes due to the location of the signalised intersection which is near to Jawi toll plaza. The study has managed to contribute to two major findings at the traffic operations at toll plaza. The first contribution is on the prediction of traffic operation at the toll plaza in the future after the implementation of full electronic toll collection system at conventional toll plazas. The second contribution is on the estimation of the actual capacity of the conventional toll plazas

A PTAS for the Highway Problem

In the highway problem, we are given an n-edge line graph (the highway), and a set of paths (the drivers), each one with its own budget. For a given assignment of edge weights (the tolls), the highway owner collects from each driver the weight of the associated path, when it does not exceed the budget of the driver, and zero otherwise. The goal is choosing weights so as to maximize the profit. A lot of research has been devoted to this apparently simple problem. The highway problem was shown to be strongly NP-hard only recently [Elbassioni,Raman,Ray,Sitters-'09]. The best-known approximation is O(log n / log log n) [Gamzu,Segev-'10], which improves on the previous-best O(log n) approximation [Balcan,Blum-'06]. Better approximations are known for a number of special cases. Finding a constant (or better!) approximation algorithm for the general case is a challenging open problem. In this paper we present a PTAS for the highway problem, hence closing the complexity status of the problem. Our result is based on a novel randomized dissection approach, which has some points in common with Arora's quadtree dissection for Euclidean network design [Arora-'98]. The basic idea is enclosing the highway in a bounding path, such that both the size of the bounding path and the position of the highway in it are random variables. Then we consider a recursive O(1)-ary dissection of the bounding path, in subpaths of uniform optimal weight. Since the optimal weights are unknown, we construct the dissection in a bottom-up fashion via dynamic programming, while computing the approximate solution at the same time. Our algorithm can be easily derandomized. The same basic approach provides PTASs also for two generalizations of the problem: the tollbooth problem with a constant number of leaves and the \emph{maximum-feasibility subsystem} problem on interval matrices. In both cases the previous best approximation factors are polylogarithmic [Gamzu,Segev-'10,Elbassioni,Raman,Ray,Sitters-'09]

On Revenue Maximization with Sharp Multi-Unit Demands

We consider markets consisting of a set of indivisible items, and buyers that have {\em sharp} multi-unit demand. This means that each buyer $i$ wants a specific number $d_i$ of items; a bundle of size less than $d_i$ has no value, while a bundle of size greater than $d_i$ is worth no more than the most valued $d_i$ items (valuations being additive). We consider the objective of setting prices and allocations in order to maximize the total revenue of the market maker. The pricing problem with sharp multi-unit demand buyers has a number of properties that the unit-demand model does not possess, and is an important question in algorithmic pricing. We consider the problem of computing a revenue maximizing solution for two solution concepts: competitive equilibrium and envy-free pricing. For unrestricted valuations, these problems are NP-complete; we focus on a realistic special case of "correlated values" where each buyer $i$ has a valuation v_i\qual_j for item $j$, where $v_i$ and \qual_j are positive quantities associated with buyer $i$ and item $j$ respectively. We present a polynomial time algorithm to solve the revenue-maximizing competitive equilibrium problem. For envy-free pricing, if the demand of each buyer is bounded by a constant, a revenue maximizing solution can be found efficiently; the general demand case is shown to be NP-hard.Comment: page2

Independent Set, Induced Matching, and Pricing: Connections and Tight (Subexponential Time) Approximation Hardnesses

We present a series of almost settled inapproximability results for three fundamental problems. The first in our series is the subexponential-time inapproximability of the maximum independent set problem, a question studied in the area of parameterized complexity. The second is the hardness of approximating the maximum induced matching problem on bounded-degree bipartite graphs. The last in our series is the tight hardness of approximating the k-hypergraph pricing problem, a fundamental problem arising from the area of algorithmic game theory. In particular, assuming the Exponential Time Hypothesis, our two main results are: - For any r larger than some constant, any r-approximation algorithm for the maximum independent set problem must run in at least 2^{n^{1-\epsilon}/r^{1+\epsilon}} time. This nearly matches the upper bound of 2^{n/r} (Cygan et al., 2008). It also improves some hardness results in the domain of parameterized complexity (e.g., Escoffier et al., 2012 and Chitnis et al., 2013) - For any k larger than some constant, there is no polynomial time min (k^{1-\epsilon}, n^{1/2-\epsilon})-approximation algorithm for the k-hypergraph pricing problem, where n is the number of vertices in an input graph. This almost matches the upper bound of min (O(k), \tilde O(\sqrt{n})) (by Balcan and Blum, 2007 and an algorithm in this paper). We note an interesting fact that, in contrast to n^{1/2-\epsilon} hardness for polynomial-time algorithms, the k-hypergraph pricing problem admits n^{\delta} approximation for any \delta >0 in quasi-polynomial time. This puts this problem in a rare approximability class in which approximability thresholds can be improved significantly by allowing algorithms to run in quasi-polynomial time.Comment: The full version of FOCS 201