  This paper addresses the minmax regret 1-sink location problem on dynamic
flow path networks with parametric weights. We are given a dynamic flow network
consisting of an undirected path with positive edge lengths, positive edge
capacities, and nonnegative vertex weights. A path can be considered as a road,
an edge length as the distance along the road and a vertex weight as the number
of people at the site. An edge capacity limits the number of people that can
enter the edge per unit time. We consider the problem of locating a sink in the
network, to which all the people evacuate from the vertices as quickly as
possible. In our model, each weight is represented by a linear function in a
common parameter $t$, and the decision maker who determines the location of a
sink does not know the value of $t$. We formulate the sink location problem
under such uncertainty as the minmax regret problem. Given $t$ and a sink
location $x$, the cost of $x$ under $t$ is the sum of arrival times at $x$ for
all the people determined by $t$. The regret for $x$ under $t$ is the gap
between the cost of $x$ under $t$ and the optimal cost under $t$. The task of
the problem is formulated as the one to find a sink location that minimizes the
maximum regret over all $t$. For the problem, we propose an $O(n^4
2^{\alpha(n)} \alpha(n) \log n)$ time algorithm where $n$ is the number of
vertices in the network and $\alpha(\cdot)$ is the inverse Ackermann function.
Also for the special case in which every edge has the same capacity, we show
that the complexity can be reduced to $O(n^3 2^{\alpha(n)} \alpha(n) \log n)$.
