Path Enumeration Research Articles

A demand model with departure time choice for within-day dynamic traffic assignment

A within-day dynamic demand model is formulated, embodying, in addition to the classic generation, distribution and modal split stages, an actual demand model taking into account departure time choice. The work focuses on this last stage, represented through an extension of the discrete choice framework to a continuous choice set. The dynamic multimodal supply and equilibrium model based on implicit path enumeration, which have been developed in previous work are outlined here, to define within-day dynamic elastic demand stochastic multimodal equilibrium as a fixed point problem on users flows and transit line frequencies. A MSA algorithm capable, in the case of Logit route choice models, of supplying equilibrium flows and frequencies on real dimension networks, is presented, as well as the specific procedures implementing the departure time choice and actual demand models. Finally, the results obtained on a test network are presented and conclusions are drawn.

Matchings Avoiding Partial Patterns and Lattice Paths

In this paper, we consider matchings avoiding partial patterns $1123$ and $1132$. We give a bijection between $1123$-avoiding matchings with $n$ edges and nonnegative lattice paths from $(0,2)$ to $(2n,0)$. As a consequence, the refined enumeration of $1123$-avoiding matchings can be reduced to the enumeration of certain lattice paths. Another result of this paper is a bijection between $1132$-avoiding matchings with $n$ edges and lattice paths from $(0,0)$ to $(2n,0)$ starting with an up step, which may go under the $x$-axis.

Elliptic enumeration of nonintersecting lattice paths

We enumerate lattice paths in the planar integer lattice consisting of positively directed unit vertical and horizontal steps with respect to a specific elliptic weight function. The elliptic generating function of paths from a given starting point to a given end point evaluates to an elliptic generalization of the binomial coefficient. Convolution gives an identity equivalent to Frenkel and Turaev's V 9 10 summation. This appears to be the first combinatorial proof of the latter, and at the same time of some important degenerate cases including Jackson's ϕ 7 8 and Dougall's F 6 7 summation. By considering nonintersecting lattice paths we are led to a multivariate extension of the V 9 10 summation which turns out to be a special case of an identity originally conjectured by Warnaar, later proved by Rosengren. We conclude with discussing some future perspectives.

Applying Branch-and-Bound Technique to Route Choice Set Generation

An algorithm to solve explicitly the path enumeration problem is proposed. This algorithm is based on the branch-and-bound technique and belongs to the class of deterministic methods along with existing approaches that combine heuristic or randomization procedures with shortest-path search. The branch-and-bound algorithm is formulated, and a methodology is designed for the application of deterministic approaches to a real case study. Path sets generated with different methods are compared for behavioral consistency, namely, the ability to reproduce actual routes chosen by individuals driving habitually from home to work. Choice set compositions for modeling purposes are determined for the consistency of the path generation process with the observed behavior. Further, model estimates and performance for different route choice specifications are examined for both path set compositions. Results suggest that the proposed branch-and-bound algorithm generates realistic and heterogeneous routes, reproduces better the observed behavior of the interviewed drivers, and produces a good choice set for route choice model estimation and performance comparison.

Applying Branch-and-Bound Technique to Route Choice Set Generation

An algorithm to solve explicitly the path enumeration problem is proposed. This algorithm is based on the branch-and-bound technique and belongs to the class of deterministic methods along with existing approaches that combine heuristic or randomization procedures with shortest-path search. The branch-and-bound algorithm is formulated, and a methodology is designed for the application of deterministic approaches to a real case study. Path sets generated with different methods are compared for behavioral consistency, namely, the ability to reproduce actual routes chosen by individuals driving habitually from home to work. Choice set compositions for modeling purposes are determined for the consistency of the path generation process with the observed behavior. Further, model estimates and performance for different route choice specifications are examined for both path set compositions. Results suggest that the proposed branch-and-bound algorithm generates realistic and heterogeneous routes, reproduces better the observed behavior of the interviewed drivers, and produces a good choice set for route choice model estimation and performance comparison.

Estimating Freight Transportation System Capacity, Flexibility, and Degraded-Condition Performance

This paper extends previous models of rail freight system capacity and capacity flexibility in three important ways. First, uncertainty in the future traffic pattern is included; second, volume-delay functions and a level-of-service constraint are added to represent deterioration of service quality as capacity limits in individual facilities are approached; and third, a stochastic traffic assignment procedure is added to eliminate the need for path enumeration and to make the model more useful for large networks. These enhancements make the model suitable for use in the assessment of the performance of a freight network under conditions in which individual links or terminals have degraded capacity. An illustration of the model as applied to the intermodal double-stack container network in the western United States indicates the improved estimates of capacity and capacity flexibility that the model provides.

Estimating Freight Transportation System Capacity, Flexibility, and Degraded-Condition Performance

This paper extends previous models of rail freight system capacity and capacity flexibility in three important ways. First, uncertainty in the future traffic pattern is included; second, volume–delay functions and a level-of-service constraint are added to represent deterioration of service quality as capacity limits in individual facilities are approached; and third, a stochastic traffic assignment procedure is added to eliminate the need for path enumeration and to make the model more useful for large networks. These enhancements make the model suitable for use in the assessment of the performance of a freight network under conditions in which individual links or terminals have degraded capacity. An illustration of the model as applied to the intermodal double-stack container network in the western United States indicates the improved estimates of capacity and capacity flexibility that the model provides.

Computational methods for determining the latest starting times and floats of tasks in interval-valued activity networks

In project management, three quantities are often used by project managers: the earliest starting date, the latest starting date and the float of tasks. These quantities are computed by the Program Evaluation and Review Techniques/Critical Path Method (PERT/CPM) algorithm. When task durations are ill known, as is often the case at the beginning of a project, they can be modeled by means of intervals, representing the possible values of these task durations. With such a representation, the earliest starting dates, the latest starting dates and the floats are also intervals. The purpose of this paper is to give efficient algorithms for their computation. After recalling the classical PERT/CPM problem, we present several properties of the concerned quantities in the interval-valued case, showing that the standard criticality analysis collapses. We propose an efficient algorithm based on path enumeration to compute optimal intervals for latest starting times and floats in the general case, and a simpler polynomial algorithm in the case of series-parallel activity networks.

The Flagged Cauchy Determinant

We consider a flagged form of the Cauchy determinant, for which we provide a combinatorial interpretation in terms of nonintersecting lattice paths. In combination with the standard determinant for the enumeration of nonintersecting lattice paths, we are able to give a new proof of the Cauchy identity for Schur functions. Moreover, by choosing different starting and end points for the lattice paths, we are led to a lattice path proof of an identity of Gessel which expresses a Cauchy-like sum of Schur functions in terms of the complete symmetric functions.

Logit-Based Stochastic User Equilibrium Problem for Entry-Exit Toll Schemes

This paper proposes a logit-based stochastic user equilibrium (SUE) problem for networks with entry-exit toll schemes that result in nonadditive path costs. First, it develops a strictly convex minimization model in terms of path flows for the problem. Second, by decomposing the original problem into two kinds of subproblems that possess the additive path costs, this study designs a novel two-stage stochastic loading algorithm exempted from path enumeration or generation. Hence, the method of successive average embedded with the proposed stochastic loading algorithm can identify the logit-based SUE link and entry-exit flows. Finally, two numerical examples are used to demonstrate the proposed model and algorithm.

Path enumeration by finding the constrained K-shortest paths

This paper deals with algorithms for finding the constrained K-shortest paths (CKSP) and their application to the path enumeration problem. An attractive property of using Constrained Shortest Paths for path enumeration is that paths can be selected based on objective criteria. The conventional way of finding these paths is to compute a sufficiently large number of overall shortest paths, and deleting the ones that do not satisfy the constraints. However for realistically sized networks, combined with restrictive constraints this method becomes unfeasible because of CPU time restrictions. A new method is proposed that finds the feasible shortest paths directly and can be applied in combination with a wide class of constraints. The paper explains how this CKSP algorithm can be implemented using the ordinary shortest path computation as its elementary operation. An example is provided in which the method is used to enumerate paths while avoiding strongly overlapping and overly circuitous paths. In this context the computational performance of the CKSP method is compared with that of the conventional method. On a network consisting of 200 nodes a speed-up factor exceeding 62 has been demonstrated on a problem that involves finding the 200 constrained shortest paths. The speed-up factor increases sharply with the size of the network and the level of restriction of the constraints. As opposed to the conventional method, the proposed implementation of the CKSP method displays only a limited sensitivity to the level of restriction of the constraints. While the conventional method could only deal with small networks, the proposed method can also enumerate paths for more realistically sized networks.

The statistic “number of udu's” in Dyck paths

In the present paper, we consider the statistic “number of udu's” in Dyck paths. The enumeration of Dyck paths according to semilength and various other parameters has been studied in several papers. However, the statistic “number of udu's” has been considered only recently. Let D n denote the set of Dyck paths of semilength n and let T n , k , L n , k , H n , k and W n , k ( r ) denote the number of Dyck paths in D n with k udu's, with k udu's at low level, at high level, and at level r ⩾ 2 , respectively. We derive their generating functions, their recurrence relations and their explicit formulas. A new setting counted by Motzkin numbers is also obtained. Several combinatorial identities are given and other identities are conjectured.

Symbolic reliability evaluation of a software with network structure

This paper proposes a graph theoretic approach for enumeration of paths and symbolic reliability expression of a large software from the program graph having a series, parallel or complex network structure. It also presents a brief account of the factors which influence the overall reliability of a software. One space application program has been used for illustration purpose.

A within-day dynamic traffic assignment model for urban road networks

In this paper a new formulation of within-day dynamic traffic assignment is presented, where a dynamic user equilibrium is expressed as a fixed-point problem in terms of arc flow temporal profiles. Specifically, it is shown that, by extending to the dynamic case the concept of network loading map, one need not introduce the continuous network loading problem in order to ensure the temporal consistency of the supply model. On this basis it is possible to devise efficient assignment algorithms, based on piece-wise linear or piece-wise constant approximation of temporal profiles over predefined time intervals covering the period of analysis, whose complexity is equal to the one resulting in the static case multiplied by the number of time intervals. With specific reference to a Logit path choice model, an implicit path enumeration network loading procedure is obtained as an extension of Dial's algorithm; then, the fixed-point problem is solved through the Bather's method.

Agent-Based Approach to Travel Demand Modeling: Exploratory Analysis

An agent-based travel demand model is developed in which travel demand emerges from the interactions of three types of agents in the transportation system: node, are, and traveler. Simple local rules of agent behaviors are shown to be capable of efficiently solving complicated transportation problems such as trip distribution and traffic assignment. A unique feature of the agent-based model is that it explicitly models the goal, knowledge, searching behavior, and learning ability of related agents. The proposed model distributes trips from origins to destinations in a disaggregate manner and does not require path enumeration or any standard shortest-path algorithm to assign traffic to the links. A sample 10-by-10 grid network is used to facilitate the presentation. The model is also appfied to the Chicago, Illinois, sketch transportation network with nearly 1,000 trip generators and sinks, and possible cafibration procedures are discussed. Agent-based modeling techniques provide a flexible travel forecasting framework that facilitates the prediction of important macroscopic travel patterns from microscopic agent behaviors and hence encourages studies on individual travel behaviors. Future research directions are identified, as is the relationship between the agent-based and activity-based approaches for travel forecasting.

Exchange relations, Dyck paths and copolymer adsorption

We consider a lattice model of fully directed copolymer adsorption equivalent to the enumeration of vertex-coloured Dyck paths. For two infinite families of periodic colourings we are able to solve the model exactly using a type of symmetry we call an exchange relation. For one of these families we are able to find an asymptotic expression for the location of the critical adsorption point as a function of the period of the colouring. This expression describes the effect of a regular inhomogeneity in the polymer on the adsorption transition.

Fractal and statistical properties of large compact polymers: a computational study

We propose a novel combinatorial algorithm for efficient generation of Hamiltonian walks and cycles on a cubic lattice, modeling the conformations of lattice toy proteins. Through extensive tests on small lattices (allowing complete enumeration of Hamiltonian paths), we establish that the new algorithm, although not perfect, is a significant improvement over the earlier approach by Ramakrishnan et al. [J Chem Phys 103(17) 7592 (1995)], as it generates the sample of conformations with dramatically reduced statistical bias. Using this method, we examine the fractal properties of typical compact conformations. In accordance with Flory theorem celebrated in polymer physics, chain pieces are found to follow Gaussian statistics on the scale smaller than the globule size. Cross-over to this Gaussian regime is found to happen at the scales which are numerically somewhat larger than previously believed. We further used Alexander and Vassiliev degrees 2 and 3 topological invariants to identify the trivial knots among the Hamiltonian loops. We found that the probability of being knotted increases with loop length much faster than it was previously thought, and that chain pieces are consistently more compact than Gaussian if the global loop topology is that of a trivial knot.

Modeling private highways in networks with entry–exit based toll charges

Previous studies on private highways generally involve network equilibrium models with link-specific and hence link-additive toll charges. In reality, toll charges for private highways depend on the entry and exit points, which are not always link-additive. This study formulates and solves the optimal toll design problem of private highways with entry–exit based toll charges using a bi-level programming approach. The lower-level traffic equilibrium problem with entry–exit based toll charges is still formulated as an optimization problem and the Frank–Wolfe algorithm is adapted for finding its solution, where the descent direction-finding sub-problem (all-or-nothing traffic assignment) is solved via a simple network transformation. The proposed method circumvents the difficulty of path enumeration or generation frequently involved in general non-additive traffic assignment problems and, hence, has the potentials to efficiently solve large network problems. Following an exploration of the properties of the lower-level traffic equilibrium sub-problem, the bi-level optimal toll design problem is solved by a recently developed efficient marginal function approach.

The use of game theory to measure the vulnerability of stochastic networks

Conventional approaches to network reliability analysis are based on either connectivity or capacity. This paper proposes an alternative method which seeks to identify those links or nodes whose failure would impair network performance the most. It is assumed that all links have two costs, a normal cost and a failed cost, both of which can be traffic-dependent. A 2-player, noncooperative, zero-sum game is envisaged between a router, seeking a least-cost path, and a virtual network tester, seeking to maximize trip-cost by failing 1 link. At the mixed strategy Nash equilibrium, link-use probabilities are optimal for the router, and link-failure probabilities are optimal for the tester. Finding the equilibrium involves solving a maximin programming problem. When link costs are fixed (not traffic-dependent), the maximin problem can be recast as a linear programming problem. Two forms of the linear programming problem are presented, one requiring path enumeration, and the other not. The interpretation of the primal and dual variables is elucidated by two propositions. Where link costs are traffic-dependent (e.g., where queuing is a feature), the mixed strategy Nash equilibrium can be found by the VISA (method of successive averages). A numerical example illustrates the approach on a stochastic network with queuing. While the example relates to single commodity e.g., where there are multiple origins and destinations.

A type-B associahedron

The (type-A) associahedron is a polytope related to polygon dissections which arises in several mathematical subjects. We propose a B-analogue of the associahedron. Our original motivation was to extend the analogies between type-A and type-B noncrossing partitions, by exhibiting a simplicial polytope whose h-vector is given by the rank-sizes of the type-B noncrossing partition lattice, just as the h-vector of the (simplicial type-A) associahedron is given by the Narayana numbers. The desired polytope Q B n is constructed via stellar subdivisions of a simplex, similarly to Lee's construction of the associahedron. As in the case of the (type-A) associahedron, the faces of Q B n can be described in terms of dissections of a convex polygon, and the f-vector can be computed from lattice path enumeration. Properties of the simple dual Q B∗ n are also discussed and the construction of a space tessellated by Q B∗ n is given. Additional analogies and relations with type A and further questions are also discussed.