Network Design on Undirected Series-Parallel Graphs
Ishan Bansal, Ryan Mao, Avhan Mishra
TL;DR
The paper addresses $BCMFP$ and $CapNDP$ on undirected series–parallel graphs with all-or-nothing edge costs, where the source–sink pair need not align with graph terminals. It develops a pseudopolynomial dynamic program that leverages the series–parallel decomposition and circulations to solve these problems exactly in that regime, and then derives an $FPTAS$ for $BCMFP$ via capacity scaling. It also identifies polynomial-time solvable special cases, such as capacities from a fixed lattice and a novel edge-upgrade gadget that preserves the SP structure, while noting the absence of a known $FPTAS$ for $CapNDP$ in this setting. Overall, the work extends the algorithmic toolkit for undirected SP graphs, highlighting structural differences from directed cases and offering practical approximation approaches for network design under budgets and demand constraints.
Abstract
We study the single pair capacitated network design problem and the budget constrained max flow problem on undirected series-parallel graphs. These problems were well studied on directed series-parallel graphs, but little is known in the context of undirected graphs. The major difference between the cases is that the source and sink of the problem instance do not necessarily coincide with the terminals of the underlying series-parallel graph in the undirected case, thus creating certain complications. We provide pseudopolynomial time algorithms to solve both of the problems and provide an FPTAS for the budget constrained max flow problem. We also provide some extensions, arguing important cases when the problems are polynomial-time solvable, and describing a series-parallel gadget that captures an edge upgrade version of the problems.