Routing multi-class traffic flows in the plane
چکیده
We study a class of multi-commodity flow problems in geometric domains: For a given planar domain P populated with obstacles (holes) of K ≥ 2 types, compute a set of thick paths from a “source” edge of P to a “sink” edge of P for vehicles of K distinct classes. Each class k of vehicle has a given set, Ok, of obstacles it must avoid and a certain width, wk, of path it requires. The problem is to determine if it is possible to route Nk width-wk paths for class k vehicles from source to sink, with each path avoiding the requisite set Ok of obstacles, and no two paths overlapping. This form of multi-commodity flow in two-dimensional domains arises in computing throughput capacity for multiple classes of aircraft in an airspace impacted by different types of constraints, such as those arising from weather hazards. We give both algorithmic theory results and experimental results. We show hardness of many versions of the problem by proving that two simple variants are NP-hard even in the case K = 2. If w1 = w2 = 1, then the problem is NP-hard even when O1 = ∅. If w1 = 2, w2 = 3, then the problem is NP-hard even when O1 = O2. In contrast, the problem for a single width and a single type of obstacles is polynomially solvable. We present approximation algorithms for the multi-criteria optimization problems that arise when trying to maximize the number of routable paths. We also give a polynomial-time algorithm for the case in which the number of holes in the input domain is bounded. Finally, we give experimental results based on an implementation of our methods and experiment with enhanced heuristics for efficient solutions in practice. Our algorithms are being utilized in simulations with NASA’s Future Air traffic management Concepts Evaluation Tool (FACET). We report on experimental results based on applying our algorithms to weather-impacted airspaces, comparing heuristic strategies for searching for feasible path orderings and for computing short multi-class routes. Our results show that multi-class routes can feasibly be computed on real weather data instances on the scale required in air traffic management applications. ∗This work grew out of discussions during our participation in the ATM-Wx Impact Modeling Workshop 2: ATM Weather Constraint Modeling, at the National Center for Atmospheric Research, Boulder, CO, March 3-4, 2008. We thank William Chan (NASA Ames), Nathan Downs (Metron Aviation), Jimmy Krozel (Metron Aviation), Tenny Lindholm (STAR), Joseph Prete (Metron Aviation), and Bob Sharman (STAR) for their input in formulating the algorithmic model and providing aviation expertise and weather data for our experiments. We thank Nathan Downs (Metron Aviation) for assistance in FACET simulations. FACET is provided by NASA Ames. We thank Esther Arkin, Petteri Kaski, Jukka Suomela, Girish Sabhnani and Charles Ward for helpful discussions. We thank the anonymous reviewers for their suggestions that improved the presentation of the results. This work was partially funded by NASA Ames, Metron Aviation, the National Science Foundation (CCF-0431030, CCF-0528209, CCF-0729019), and Academy of Finland (grant 138520). Preliminary version appeared in the abstracts of the 18th Annual Fall Workshop on Computational Geometry, 2008. †Google Inc., jdkim,jasonzou@google.com ‡Department of Applied Mathematics and Statistics, Stony Brook University, jsbm@ams.sunysb.edu §Helsinki Institute for Information Technology, Department of Computer Science, University of Helsinki, polishch@hiit.fi ¶Department of Computer Science, Stony Brook University, syang@cs.sunysb.edu