A Statistical Approach to the TSP

This paper is an example of the growing interface between statistics and mathematical optimization. A very efficient heuristic algorithm for the combinatorially intractable TSP is presented, from which statistical estimates of the optimal tour length can be derived. Assumptions, along with computational experience and conclusions are discussed.Supported in part by the U.S. Department of Transportation under contract DOT-TSC-1058, Transportation Advanced Research Program (TARP)

A Deep-Learning Based Framework for Joint Downlink Precoding and CSI Sparsification

Optimal pilot design to acquire channel state information (CSI) is of critical importance for FDD downlink massive MIMO systems, and is still an open problem. To tackle this issue, in this paper we propose a two-stage precoding approach based on reduced CSI (rCSI-TSP) design framework and an efficient algorithm, whose core is to obtain an optimal precoder while also sparsifying physical CSI (pCSI), so as to save on CSI estimation. The advantages of the rCSI-TSP framework are three-fold. First, the framework enables to simultaneously extract and exploit statistical and instantaneous CSI. Second, it guarantees the most needed rCSI can be obtained and thus avoids performance loss due to heuristic pilot design. Third, we tailor an efficient online deep-learning based method for the TSP framework, which paves the way for practical applications. As an example, we apply the framework to the multi-user symbol-level precoding (SLP) and verify performance improvements

A New Methodology for Evaluating the Effectiveness of Bus Rapid Transit Strategies

Over the last few years, public transportation has become more desirable as capacity of existing roadways failed to keep up with rapidly increasing traffic demand. Buses are one of the most common modes of public transportation with low impact on network capacity, especially in small and congested urban areas. However, the use of regularly scheduled buses as the main public transport mode can become useless with the presence of traffic congestion and dense construction areas. In cases like these, innovative solutions, such as bus rapid transit (BRT), can provide an increased level of service without having to resort to other, more expensive modes, such as light rail transit (LRT) and metro systems (subways). Transit signal priority (TSP), which provides priority to approaching buses at signalized intersections by extending the green or truncating the red, can also increase the performance of the bus service. Understanding the combined impact of TSP and BRT on network traffic operations can be complex. Although TSP has been implemented worldwide, none of the previous studies have examined in depth the effects of using conditional and unconditional TSP strategies with a BRT system. The objective of this research is to evaluate the effectiveness of BRT without TSP, then with conditional or unconditional TSP strategies. The micro-simulation software VISSIM was used to compare different TSP and BRT scenarios. These simulation scenarios include the base scenario (before implementation of the TSP and BRT systems), Unconditional TSP (TSP activates for all buses), Conditional TSP 3 minutes behind (TSP only activates for buses that are 3 minutes or more behind schedule), Conditional TSP 5 minutes behind (only activates for buses 5 minutes or more behind schedule), BRT with no TSP, BRT with Unconditional TSP, BRT with Conditional TSP 3 minutes behind, and BRT with Conditional TSP 5 minutes behind. The VISSIM simulation model was developed, calibrated and validated using a variety of data that was collected in the field. These data included geometric data, (number of lanes, intersection geometries, etc.); traffic data (average daily traffic volumes at major intersections, turning movement percentages at intersections, heavy vehicle percentages, bus passenger data, etc.); and traffic control data (signal types, timings and phasings, split history, etc.). Using this field data ensured the simulation model was sufficient for modeling the test corridor. From this model, the main performance parameters (for all vehicles and for buses only) for through movements in both directions (eastbound and westbound) along the corridor were analyzed for the various BRT/TSP scenarios. These parameters included average travel times, average speed profiles, average delays, and average number of stops. As part of a holistic approach, the effects of BRT and TSP on crossing street delay were also evaluated. Simulation results showed that TSP and BRT scenarios were effective in reducing travel times (up to 26 %) and delays (up to 64%), as well as increasing the speed (up to 47%), compared to the base scenario. The most effective scenarios were achieved by combining BRT and TSP. Results also showed that BRT with Conditional TSP 3 minutes behind significantly improved travel times (17 – 26%), average speed (30 – 39%), and average total delay per vehicle (11 – 32%) for the main corridor through movements compared with the base scenario, with only minor effects on crossing street delays. BRT with Unconditional TSP resulted in significant crossing street delays, especially at major intersections with high traffic demand, which indicates that this scenario is impractical for implementation in the corridor. Additionally, BRT with Conditional TSP 3 minutes behind had better travel time savings than BRT with Conditional TSP 5 minutes behind for both travel directions, making this the most beneficial scenario. This research provided an innovative approach by using nested sets (hierarchical design) of TSP and BRT combination scenarios. Coupled with microscopic simulation, nested sets in the hierarchical design are used to evaluate the effectiveness of BRT without TSP, then with conditional or unconditional TSP strategies. The robust methodology developed in this research can be applied to any corridor to understand the combined TSP and BRT effects on traffic performance. Presenting the results in an organized fashion like this can be helpful in decision making. This research investigated the effects of BRT along I-Drive corridor (before and after conditions) at the intersection level. Intersection analysis demonstrated based on real life data for the before and after the construction of BRT using the Highway Capacity SoftwareTM (HCS2010) that was built based on the Highway Capacity Manual (HCM 2010) procedures for urban streets and signalized intersections. The performance measure used in this analysis is the level of service (LOS) criteria which depends on the control delay (seconds per vehicle) for each approach and for the entire intersection. The results show that implementing BRT did not change the LOS. However, the control delay has improved at most of the intersections\u27 approaches. The majority of intersections operated with an overall LOS C or better except for Kirkman Road intersection (T2) with LOS E because it has the highest traffic volumes before and after BRT construction. This research also used regression analysis to observe the effect of the tested scenarios analyzed in VISSIM software compared to the No TSP – No BRT base model for all vehicles and for buses only. The developed regression model can predict the effect of each scenario on each studied Measures of Performance (MOE). Minitab statistical software was used to conduct this multiple regression analysis. The developed models with real life data input are able to predict how proposed enhancements change the studied MOEs. The BRT models presented in this research can be used for further sensitivity analysis on a larger regional network in the upcoming regional expansion of the transit system in Central Florida. Since this research demonstrated the operational functionality and effectiveness of BRT and TSP systems in this critical corridor in Central Florida, these systems\u27 accomplishments can be expanded throughout the state of Florida to provide greater benefits to transit passengers. Furthermore, to demonstrate the methodology developed in this research, it is applied to a test corridor along International Drive (I-Drive) in Orlando, Florida. This corridor is key for regional economic prosperity of Central Florida and the novel approach developed in this dissertation can be expanded to other transit systems

An Evolutionary Strategy based on Partial Imitation for Solving Optimization Problems

In this work we introduce an evolutionary strategy to solve combinatorial optimization tasks, i.e. problems characterized by a discrete search space. In particular, we focus on the Traveling Salesman Problem (TSP), i.e. a famous problem whose search space grows exponentially, increasing the number of cities, up to becoming NP-hard. The solutions of the TSP can be codified by arrays of cities, and can be evaluated by fitness, computed according to a cost function (e.g. the length of a path). Our method is based on the evolution of an agent population by means of an imitative mechanism, we define `partial imitation'. In particular, agents receive a random solution and then, interacting among themselves, may imitate the solutions of agents with a higher fitness. Since the imitation mechanism is only partial, agents copy only one entry (randomly chosen) of another array (i.e. solution). In doing so, the population converges towards a shared solution, behaving like a spin system undergoing a cooling process, i.e. driven towards an ordered phase. We highlight that the adopted `partial imitation' mechanism allows the population to generate solutions over time, before reaching the final equilibrium. Results of numerical simulations show that our method is able to find, in a finite time, both optimal and suboptimal solutions, depending on the size of the considered search space.Comment: 18 pages, 6 figure

The random link approximation for the Euclidean traveling salesman problem

The traveling salesman problem (TSP) consists of finding the length of the shortest closed tour visiting N ``cities''. We consider the Euclidean TSP where the cities are distributed randomly and independently in a d-dimensional unit hypercube. Working with periodic boundary conditions and inspired by a remarkable universality in the kth nearest neighbor distribution, we find for the average optimum tour length = beta_E(d) N^{1-1/d} [1+O(1/N)] with beta_E(2) = 0.7120 +- 0.0002 and beta_E(3) = 0.6979 +- 0.0002. We then derive analytical predictions for these quantities using the random link approximation, where the lengths between cities are taken as independent random variables. From the ``cavity'' equations developed by Krauth, Mezard and Parisi, we calculate the associated random link values beta_RL(d). For d=1,2,3, numerical results show that the random link approximation is a good one, with a discrepancy of less than 2.1% between beta_E(d) and beta_RL(d). For large d, we argue that the approximation is exact up to O(1/d^2) and give a conjecture for beta_E(d), in terms of a power series in 1/d, specifying both leading and subleading coefficients.Comment: 29 pages, 6 figures; formatting and typos correcte

Computational Complexity for Physicists

These lecture notes are an informal introduction to the theory of computational complexity and its links to quantum computing and statistical mechanics.Comment: references updated, reprint available from http://itp.nat.uni-magdeburg.de/~mertens/papers/complexity.shtm

Scaling and Universality in Continuous Length Combinatorial Optimization

We consider combinatorial optimization problems defined over random ensembles, and study how solution cost increases when the optimal solution undergoes a small perturbation delta. For the minimum spanning tree, the increase in cost scales as delta^2; for the mean-field and Euclidean minimum matching and traveling salesman problems in dimension d>=2, the increase scales as delta^3; this is observed in Monte Carlo simulations in d=2,3,4 and in theoretical analysis of a mean-field model. We speculate that the scaling exponent could serve to classify combinatorial optimization problems into a small number of distinct categories, similar to universality classes in statistical physics.Comment: 5 pages; 3 figure