# Minmax Regret *k*-Sink Location on a Dynamic Path Network with Uniform Capacities

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## Abstract

In *Dynamic flow networks* an edge’s capacity is the amount of flow (items) that can enter it in unit time. All flow must be moved to sinks and *congestion* occurs if flow has to wait at a vertex for other flow to leave. In the *uniform capacity* variant all edges have the same capacity. The minmax *k*-sink location problem is to place *k* sinks minimizing the maximum time before all items initially on vertices can be evacuated to a sink. The *minmax regret* version introduces uncertainty into the input; the amount at each source is now only specified to within a range. The problem is to find a universal solution (placement of *k* sinks) whose regret (difference from optimal for a given input) is minimized over all inputs consistent with the given range restrictions. The previous best algorithms for the minmax regret version of the *k*-sink location problem on a path with uniform capacities ran in *O*(*n*) time for \(k=1\), \(O(n \log ^ 4 n)\) time for \(k=2\) and \({\varOmega }(n^{k+1})\) for \( k >2\). A major bottleneck to improving those solutions was that minmax regret seemed an inherently *global* property. This paper derives new combinatorial insights that allow decomposition into *local* problems. This permits designing two new algorithms. The first runs in \(O(n^3 \log n)\) time independent of *k* and the second in \(O( n k^2 \log ^{k+1} n)\) time. These improve all previous algorithms for \(k >1\) and, for \(k > 2\), are the first polynomial time algorithms known.

## Keywords

Dynamic flow networks Sink location problems Evacuation Minmax regret## Notes

## Supplementary material

## References

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