Hump Yard Track Allocation with Temporary Car Storage

In rail freight operation, freight cars need to be separated and reformed into new trains at hump yards. The classification procedure is complex and hump yards constitute bottlenecks in the rail freight network, often causing outbound trains to be delayed. One of the problems is that planning for the allocation of tracks at hump yards is difficult, given that the planner has limited resources (tracks, shunting engines, etc.) and needs to foresee the future capacity requirements when planning for the current inbound trains. In this paper, we consider the problem of allocating classification tracks in a rail freight hump yard for arriving and departing trains with predetermined arrival and departure times. The core problem can be formulated as a special list coloring problem. We focus on an extension where individual cars can temporarily be stored on a special subset of the tracks. An extension where individual cars can temporarily be stored on a special subset of the tracks is also considered. We model the problem using mixed integer programming, and also propose several heuristics that can quickly give feasible track allocations. As a case study, we consider a real-world problem instance from the Hallsberg Rangerbangård hump yard in Sweden. Planning over horizons over two to four days, we obtain feasible solutions from both the exact and heuristic approaches that allow all outgoing trains to leave on time

The Price of Anarchy in Network Creation Games Is (Mostly) Constant

We study the price of anarchy and the structure of equilibria in network creation games. A network creation game is played by n players {1,2, ,n}, each identified with a vertex of a graph (network), where the strategy of player i, i=1, ,n, is to build some edges adjacent to i. The cost of building an edge is α>0, a fixed parameter of the game. The goal of every player is to minimize its creation cost plus its usage cost. The creation cost of player i is α times the number of built edges. In the SumGame variant, the usage cost of player i is the sum of distances from i to every node of the resulting graph. In the MaxGame variant, the usage cost is the eccentricity of i in the resulting graph of the game. In this paper we improve previously known bounds on the price of anarchy of the game (of both variants) for various ranges of α, and give new insights into the structure of equilibria for various values of α. The two main results of the paper show that for α>273⋅n all equilibria in SumGame are trees and thus the price of anarchy is constant, and that for α>129 all equilibria in MaxGame are trees and the price of anarchy is constant. For SumGame this answers (almost completely) one of the fundamental open problems in the field—is price of anarchy of the network creation game constant for all values of α?—in an affirmative way, up to a tiny range of

Track Allocation in Freight-Train Classification with Mixed Tracks

We consider the process of forming outbound trains from cars of inbound trains at rail-freight hump yards. Given the arrival and departure times as well as the composition of the trains, we study the problem of allocating classification tracks to outbound trains such that every outbound train can be built on a separate classification track. We observe that the core problem can be formulated as a special list coloring problem in interval graphs, which is known to be NP-complete. We focus on an extension where individual cars of different trains can temporarily be stored on a special subset of the tracks. This problem induces several new variants of the list-coloring problem, in which the given intervals can be shortened by cutting off a prefix of the interval. We show that in case of uniform and sufficient track lengths, the corresponding coloring problem can be solved in polynomial time, if the goal is to minimize the total cost associated with cutting off prefixes of the intervals. Based on these results, we devise two heuristics as well as an integer program to tackle the problem. As a case study, we consider a real-world problem instance from the Hallsberg Rangerbangård hump yard in Sweden. Planning over horizons of seven days, we obtain feasible solutions from the integer program in all scenarios, and from the heuristics in most scenarios

Optimized shunting with mixed-usage tracks

We consider the planning of railway freight classification at hump yards, where the problem involves the formation of departing freight train blocks from arriving trains subject to scheduling and capacity constraints. The hump yard layout considered consists of arrival tracks of sufficient length at an arrival yard, a hump, classification tracks of non-uniform and possibly non-sufficient length at a classification yard, and departure tracks of sufficient length. To increase yard capacity, freight cars arriving early can be stored temporarily on specific mixed-usage tracks. The entire hump yard planning process is covered in this paper, and heuristics for arrival and departure track assignment, as well as hump scheduling, have been included to provide the neccessary input data. However, the central problem considered is the classification track allocation problem. This problem has previously been modeled using direct mixed integer programming models, but this approach did not yield lower bounds of sufficient quality to prove optimality. Later attempts focused on a column generation approach based on branch-and-price that could solve problem instances of industrial size. Building upon the column generation approach we introduce a direct arc-based integer programming model, where the arcs are precedence relations between blocks on the same classification track. Further, the most promising models are adapted for rolling-horizon planning. We evaluate the methods on historical data from the Hallsberg shunting yard in Sweden. The results show that the new arc-based model performs as well as the column generation approach. It returns an optimal schedule within the execution time limit for all instances but from one, and executes as fast as the column generation approach. Further, the short execution times of the column generation approach and the arc-indexed model make them suitable for rolling-horizon planning, while the direct mixed integer program proved to be too slow for this. Extended analysis of the results shows that mixing was only required if the maximum number of concurrent trains on the classification yard exceeds 29 (there are 32 available tracks), and that after this point the number of extra car roll-ins increases heavily

Mapping Simple Polygons: How Robots Benefit from Looking Back

We consider the problem of mapping an initially unknown polygon of size n with a simple robot that moves inside the polygon along straight lines between the vertices. The robot sees distant vertices in counter-clockwise order and is able to recognize the vertex among them which it came from in its last move, i.e.the robot can look back. Other than that the robot has no means of distinguishing distant vertices. We assume that an upper bound on n is known to the robot beforehand and show that it can always uniquely reconstruct the visibility graph of the polygon. Additionally, we show that multiple identical and deterministic robots can always solve the weak rendezvous problem in which the robots need to position themselves such that all of them are mutually visible to each other. Our results are tight in the sense that the strong rendezvous problem, where robots need to gather at a vertex, cannot be solved in general, and, without knowing a bound beforehand, not even n can be determined. In terms of mobile agents exploring a graph, our result implies that they can reconstruct any graph that is the visibility graph of a simple polygon. This is in contrast to the known result that the reconstruction of arbitrary graphs is impossible in general, even if n is know

How to Guard a Graph?

We initiate the study of the algorithmic foundations of games in which a set of cops has to guard a region in a graph (or digraph) against a robber. The robber and the cops are placed on vertices of the graph; they take turns in moving to adjacent vertices (or staying). The goal of the robber is to enter the guarded region at a vertex with no cop on it. The problem is to find the minimum number of cops needed to prevent the robber from entering the guarded region. The problem is highly non-trivial even if the robber's or the cops' regions are restricted to very simple graphs. The computational complexity of the problem depends heavily on the chosen restriction. In particular, if the robber's region is only a path, then the problem can be solved in polynomial time. When the robber moves in a tree (or even in a star), then the decision version of the problem is NP-complete. Furthermore, if the robber is moving in a directed acyclic graph, the problem becomes PSPACE-complet

Robust routing in urban public transportation: How to find reliable journeys based on past observations.

Abstract We study the problem of robust routing in urban public transportation networks. In order to propose solutions that are robust for typical delays, we assume that we have past observations of real traffic situations available. In particular, we assume that we have "daily records" containing the observed travel times in the whole network for a few past days. We introduce a new concept to express a solution that is feasible in any record of a given public transportation network. We adapt the method of Buhmann et al. ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems, G.2.2 Graph Theory (Graph algorithms, Network problems), I.2.6 Learning Keywords and phrases Introduction We study the problem of routing in urban public transportation networks, such as tram and bus networks in large cities, focusing on the omnipresent uncertain situations when (typical) delays occur. In particular, we search for robust routes that allow reliable yet quick passenger transportation. We think of a "dense" tram network in a large city containing many tram lines, where each tram line is a sequence of stops that is served repeatedly during the day, and where there are several options to get from one location to another. Such a network usually does not contain clear hierarchical structure (as opposed to train networks), and each line is served with high frequency. Given two tram stops a and b together with a latest arrival time t A , our goal is to provide a simple yet robust description of how to travel in the

Computing and Listing st-Paths in Subway Networks

International audienceGiven a set of paths (called lines) L, a subway network is a graph GL = (V, A) where V contains exactly the vertices and arcs of every line l ∈ L. An st-route is a pair (π, γ) where γ = (l1,. .. , l h) is a line sequence and π is an st-path in GL which is the concatena-tion of subpaths of the lines l1,. .. , l h , in this order. We study three related problems concerning traveling from s to t in GL. We present an efficient (i.e., polynomial-time) algorithm for computing an st-route (π, γ) where |γ| (i.e., the number of line changes plus one) is minimum among all st-routes. We show for the problem of finding an st-route (π, γ) that minimizes the number of different lines in γ, even computing an o(log |V |)-approximation is NP-hard. Finally, given an integer β, we present an algorithm for enumerating all st-paths π for which a route (π, γ) with |γ| ≤ β exists, and show that the running time of this algorithm is polynomial with respect to both input and output size

A (4 + epsilon)-Approximation for the Minimum-Weight Dominating Set Problem in Unit Disk Graphs

We present a (4 + e)-approximation algorithm for the problem of computing a minimum-weight dominating set in unit disk graphs, where e is an arbitrarily small constant. The previous best known approximation ratio was 5 + e. The main result of this paper is a 4-approximation algorithm for the problem restricted to constant-size areas. To obtain the (4 + e)-approximation algorithm for the unrestricted problem, we then follow the general framework from previous constant-factor approximations for the problem: we consider the problem in constant-size areas, and combine the solutions obtained by our 4-approximation algorithm for the restricted case to get a feasible solution for the whole problem. Using the shifting technique (selecting a best solution from several considered partitionings of the problem into constant-size areas) we obtain the claimed (4 + e)-approximation algorithm. By combining our algorithm with a known algorithm for node-weighted steiner trees, we obtain a 7.875-approximation for the minimum-weight connected dominating set problem in unit disk graphs