Abstract
Road networks are a classical stage for applications in network science and graph theory. Meanwhile, many combinatorial problems that arise in road networks are computationally intractable. Thus, an attractive way of tackling them is through efficient heuristics with provable performance guarantees, better known as approximation algorithms. This motivates the intersection of algorithm design with the aforementioned fields. Specifically, identifying measures that characterize graphs and exploiting them in the design of algorithms may yield practical heuristics with rigorous mathematical justification. Herein, we propose a new graph measure, namely the asymmetry factor ΔG of a directed graph G, with immediate algorithmic results via a symmetrization procedure and the black box use of approximation algorithms for symmetric graphs. Crucially, we analyze the asymmetry factors of the road networks from a diverse set of twelve cities, providing empirical evidence that road networks exhibit low bounded asymmetry and thereby justifying the practical use of algorithms for symmetric graphs.
Highlights
Road networks are one of the classical stageas for applications in network science and graph theory
The optimization versions of these problems are referred to as NP-hard. It remains an open question whether such an algorithm exists: this is the renowned P versus NP millennium question[10], where P and NP are the classes of decision problems that can be solved and verified in polynomial time, respectively
We show that directed graphs with bounded asymmetry ΔG allow practical constant factor approximation algorithms for some discrete optimization problems such as, but not restricted to, the Asymmetric Traveling Salesman Problem (ATSP) via a symmetrization procedure and the black box use of constant factor approximation algorithms for symmetric graphs
Summary
Road networks are one of the classical stageas for applications in network science and graph theory. In 1735, Euler resolved the long-standing Bridges of Königsberg Problem, which asked whether it was possible to perform a tour through the town while visiting each bridge in a set of seven exactly once[7] He was able to solve the problem by viewing it through the lens of an abstraction; namely a set of points connected by edges representing bridges. Lincoln’s circuit tour problem is more popularly known as the Traveling Salesman Problem (TSP), whose decision version is NP-complete Despite this fact, NP-hard transportation problems in the real world are still being ‘solved’ on a regular basis. By solved we mean that we use efficient algorithms that may not always find the absolute best solution, but that for most practical purposes produce feasible solutions of reasonable quality These are referred to as heuristics, and www.nature.com/scientificreports/. Boeing[22] proposes a comprehensive typology for many of the measures above in the context of road networks
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