Network construction/restoration problems: cycles and complexity

In network construction/restoration problems introduced in Averbakh and Pereira (IIE Trans 44(8):681-694, 2012; Eur J Oper Res 244:715-729, 2015), a server (construction crew) builds edges of a given network starting from a given vertex (the depot), with a constant construction speed. The server can travel within the already constructed part of the network with a speed that is incomparably faster than the construction speed. The recovery time of a vertex is the time when the vertex becomes connected to the depot by an already constructed path. Due dates and/or weights are associated with vertices. It is required to find an optimal construction schedule that minimizes the total weighted recovery time or the maximum lateness of the vertices. Both problems are known to be polynomially solvable on trees and NP-hard on general networks. We prove that both problems are NP-hard even on so simple extensions of trees as cactuses, and discuss some polynomially solvable cases.

Automated summary

The network construction/restoration problems involve finding an optimal construction schedule to minimize the total weighted recovery time or maximum lateness of vertices, which are known to be polynomially solvable on trees but NP-hard on general networks. They are also proven to be NP-hard even on simple extensions of trees like cactuses, with some polynomially solvable cases discussed.