### An Improved Algorithm for Packing T-Paths in Inner Eulerian Networks [chapter]

Maxim A. Babenko, Kamil Salikhov, Stepan Artamonov
2012 Lecture Notes in Computer Science
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
more » ... d by Ibaraki, Karzanov, and Nagamochi, who came up with a strongly-polynomial algorithm of complex- 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) · log T + V E) time, which is unsatisfactory since a max-flow can be found in o(V E) time. For this case, we present an improved method that runs in O(φ(V, E) · log T + E log V ) time. Thus plugging in the max-flow algorithm of Dinic, we reduce the overall complexity from O(V E) to O(min(V 2/3 E, E 3/2 ) · log T ). Preliminaries Introduction Computing a maximum integer flow, i.e. a maximum packing of paths connecting a given pair of terminals subject to edge capacities, is widely regarded as a central problem in combinatorial optimization. This problem has myriads of applications, both theoretical and practical. Given a graph G = (V, E) (either directed or undirected) and arbitrary integer capacities e : E → Z + , one of the best strongly-polynomial max-flow algorithm  runs in O(V E log(V 2 /E)) time. (Hereinafter, in notation involving functions of numerical arguments or time bounds, we indicate sets for their cardinalities.) More efficient methods are known for the special case of unit capacities. The oldest one belongs to Dinic  and runs in O(min(E 3/2 , V 2/3 E)) time (as shown independently by Karzanov  and Even and Tarjan  ). Better results Supported by RFBR grants 10-01-93109 and 12-01-00864.