Minmax Regret for sink location on paths with general capacities

In dynamic flow networks, every vertex starts with items (flow) that need to be shipped to designated sinks. All edges have two associated quantities: length, the amount of time required for a particle to traverse the edge, and capacity, the number of units of flow that can enter the edge in unit time. The goal is move all flow to the sinks. A variation of the problem, modelling evacuation protocols, is to find the sink location(s) that minimize evacuation time, restricting the flow to be CONFLUENT. Solving this problem is is NP-hard on general graphs, and thus research into optimal algorithms has traditionally been restricted to special graphs such as paths, and trees. A specialized version of robust optimization is minmax REGRET, in which the input flows at the vertices are only partially defined by constraints. The goal is to find a sink location that has the minimum{ regret} over all input flows that satisfy the partially defined constraints. Regret for a fully defined input flow and a sink is defined to be the difference between the evacuation time to that sink and the optimal evacuation time. A large recent literature derives polynomial time algorithms for the minmax regret $k$-sink location problem on paths and trees under the simplifying condition that all edges have the same (uniform) capacity. This paper develops a $O(n^4 \log n)$ time algorithm for the minmax regret $1$-sink problem on paths with general (non-uniform) capacities. To the best of our knowledge, this is the first minmax regret result for dynamic flow problems in any type of graph with general capacities.