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

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.