An Improved Algorithm for Packing T-Paths in Inner Eulerian Networks
A digraph G = (V,E) with a distinguished set T ⊆ V of terminals is called inner Eulerian if for each v ∈ V − T the numbers of arcs entering and leaving v are equal. By a T-path we mean a simple directed path connecting distinct terminals with all intermediate nodes in V − T. This paper concerns the problem of finding a maximum T-path packing, i.e. a maximum collection of arc-disjoint T-paths.
A min-max relation for this problem was established by Lomonosov. The capacitated version was studied by Ibaraki, Karzanov, and Nagamochi, who came up with a strongly-polynomial algorithm of complexity O(φ(V,E) ·logT + V 2 E) (hereinafter φ(n,m) denotes the complexity of a max-flow computation in a network with n nodes and m arcs).
For unit capacities, the latter algorithm takes O(φ(V,E) ·logT + VE) time, which is unsatisfactory since a max-flow can be found in o(VE) time. For this case, we present an improved method that runs in O(φ(V,E) ·logT + E logV) time. Thus plugging in the max-flow algorithm of Dinic, we reduce the overall complexity from O(VE) to O( min (V 2/3 E, E 3/2) ·logT).
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