[Chapter 1] 1 Potentially optimal paths and route choice in networks with arc delays By Achille Fonzone 1 , Jan Dirk Schmöcker 2 and Michael G. H. Bell 1 1 Centre for Transport Studies, Imperial College London, 2 Department of Civil and Environmental Engineering, Tokyo Institute of Technology Abstract The shortest path problem has been widely studied in graph theory and its solutions have been largely used in the transport field, both in assignment and vehicle navigation applications. Solutions have been proposed under different assumptions concerning the time-dependency and the randomness of arc costs. In particular it has been shown that in networks affected by not completely predictable events travellers can minimize their expected travel times by adopting adaptive strategies, i.e. sets of paths, instead of single routes. The present paper focuses on the problem of route guidance in a network in which the travel time on each link can assume only two values, corresponding to the free flow situation and to the delayed one. Path set selection and route choice are decoupled. The problem of finding the set of the potentially optimal paths is solved implicitly by identifying the set of the potentially optimal links. Sufficient and necessary conditions for a link to be optimal do not coincide and therefore an algorithm is presented which can identify only a subset of all potentially optimal links. An approach considering risk aversion is proposed for route choice. 1. Introduction Finding a single or set of minimum paths in networks has applications in a large number of disciplines. In transport, for example, finding the shortest path (see Pallottino & Scutellà 1997 for a critical introduction) is the basis for normative as well as predictive problems such as assignment or route guidance. Of particular interest are problems when delays might occur that are random and/or time–dependent. Sources of delays are numerous including congestion, accidents or network disruptions caused by natural disasters. When trips over a transport network are affected by such random or unpredictable en–route events, travellers can minimize their expected travel times by adopting adaptive strategies instead of single paths. This is proven by Hall (1986) in the case of stochastic time–dependent