Retrieving the cheapest path of a graph with time-dependent edge weights

Question

There are many efficient algorithms for finding the shortest path in a network, like dijkstra's or bellman-ford's. But what if the weights of edges are time-dependent? I'm trying to find an efficient algorithm to find the path of graph with the lowest cost, where weights of edges depend on what time the edge is being walked through. I need to take into account how the situation changes during the path traversal after each passed edge. How can I do this? Maybe there are ways to at least try to optimize the cost to be as low as possible?

Somewhat formal definition:

Given a graph $G=(V,E)$, time-based edge cost function $w_{v_i,v_j}(t)$ and a constant time to walk through an edge $t_{v_{i},v_{j}}$, find path from starting point $v_s$ to finish point $v_f$, $P=(v_1, v_2, ..., v_n)$, where $v_1 = v_s$, $v_n = v_f$, such that $\sum_{i=1}^{n-1}w_{v_{i},v_{i+1}}(T_{P}(i)) = min$, where $T_{P}(n) = \sum_{i=1}^{n-1} t_{v_{i}, v_{i+1}}$ (time passed since the start of the path)

Basically, the graph is a collection of pedestrian roads of a city area and the cost function is a factor that a pedestrian might want to avoid at the cost of a possibly longer path, such as sun illumination (wants to walk in the shadow) or noisiness/crowdedness (wants to avoid public places).

Answer 1

This is solvable using a product construction. You construct a new graph $G'=(V',E')$ where each vertex in $V'$ has the form $\langle v,t \rangle$, to keep track of both which vertex you're at ($v$) and the current time ($t$). Then, find a shortest path in $G'$.

To learn more about this approach, for some other examples of a product construction, see, e.g., Shortest path given correct order of colours?, https://stackoverflow.com/q/5370902/781723, Restricted traversal on an undirected graph, Dijkstra's algorithm to compute shortest paths using k edges?, How hard is finding the shortest path in a graph matching a given regular language?.