Geodesic
Maximum flow in parallel networks with connected arcs
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 2, Tome 231 (2024), pp. 44-52

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The well-known problem of finding maximum flows in classical networks has many solution algorithms that have a polynomial computational complexity depending on the size of the network. In general, the problem of finding maximum flows for networks with connected arcs is NP-complete. However, among the previously studied networks with connected arcs, there are networks for which the calculation of maximum flows is feasible in a time polynomially depending on the size of the network. This work is devoted to determining the influence of the topology of networks with connected arcs on the possibility of finding maximum flows in polynomial time. In this paper, for a class of parallel networks with connected arcs, we propose a fast polynomial algorithm for finding maximum flows.
Keywords: graph, network, network flow, maximum flow, parallel network, connected arcs, networks with reachability restrictions 
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     title = {Maximum flow in parallel networks with connected arcs},
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I. M. Erusalimskyi; V. A. Skorokhodov; V. A. Rusakov. Maximum flow in parallel networks with connected arcs. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 2, Tome 231 (2024), pp. 44-52. http://geodesic.mathdoc.fr/item/INTO_2024_231_a3/