Algorithms for flows over time with scheduling costs

• Published in: Mathematical programming Vol. 192; no. 1-2; pp. 177 - 206
• Main Authors: Frascaria, Dario, Olver, Neil
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2022; Springer; Springer Nature B.V
• Subjects: Algorithms; Analysis; Calculus of Variations and Optimal Control; Optimization; Combinatorial analysis; Combinatorics; Full Length Paper; Game theory; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Optimization; Scheduling; Theoretical; Flows over time; Secondary: 05C21; 90B20; Tolls; Traffic; Primary: 90C27
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-021-01725-z

Flows over time have received substantial attention from both an optimization and (more recently) a game-theoretic perspective. In this model, each arc has an associated delay for traversing the arc, and a bound on the rate of flow entering the arc; flows are time-varying. We consider a setting which is very standard within the transportation economic literature, but has received little attention from an algorithmic perspective. The flow consists of users who are able to choose their route but also their departure time, and who desire to arrive at their destination at a particular time, incurring a scheduling cost if they arrive earlier or later. The total cost of a user is then a combination of the time they spend commuting, and the scheduling cost they incur. We present a combinatorial algorithm for the natural optimization problem, that of minimizing the average total cost of all users (i.e., maximizing the social welfare). Based on this, we also show how to set tolls so that this optimal flow is induced as an equilibrium of the underlying game.