Stackelberg Shortest Path Tree Game, Revisited

Let $G(V,E)$ be a directed graph with $n$ vertices and $m$ edges. The edges $E$ of $G$ are divided into two types: $E_F$ and $E_P$. Each edge of $E_F$ has a fixed price. The edges of $E_P$ are the priceable edges and their price is not fixed a priori. Let $r$ be a vertex of $G$. For an assignment of prices to the edges of $E_P$, the revenue is given by the following procedure: select a shortest path tree $T$ from $r$ with respect to the prices (a tree of cheapest paths); the revenue is the sum, over all priceable edges $e$, of the product of the price of $e$ and the number of vertices below $e$ in $T$. Assuming that $k=|E_P|\ge 2$ is a constant, we provide a data structure whose construction takes $O(m+n\log^{k-1} n)$ time and with the property that, when we assign prices to the edges of $E_P$, the revenue can be computed in $(\log^{k-1} n)$. Using our data structure, we save almost a linear factor when computing the optimal strategy in the Stackelberg shortest paths tree game of [D. Bil{\`o} and L. Gual{\`a} and G. Proietti and P. Widmayer. Computational aspects of a 2-Player Stackelberg shortest paths tree game. Proc. WINE 2008].