Extremal Solutions for Network Flow with Differential Constraints

René Brandenberg () and Paul Stursberg ()
Additional contact information
René Brandenberg: School of Computation, Information and Technology, Department of Mathematics
Paul Stursberg: School of Computation, Information and Technology, Department of Mathematics

Journal of Optimization Theory and Applications, 2025, vol. 207, issue 3, No 30, 31 pages

Abstract: Abstract In network flow problems, there is a well-known one-to-one relationship between extreme points of the feasibility region and trees in the associated undirected graph. The same is true for the dual differential problem. In this paper, we study a problem variant with both differential constraints and constraints on flow conservation at every node, which we call differential flow. This variant is motivated by an application in the expansion planning of energy networks. We show that all extreme points in the differential flow polytope still directly correspond to graph-theoretical structures in the underlying network, namely a generalization of spanning trees. The reverse is generally also true except in very special cases where the network parameters satisfy a set of particular equations. We furthermore show that these exceptional cases can never occur in cactus graphs and present additional, sufficient criteria for when the one-to-one correspondence between extreme points and graph-theoretical structures holds. Finally, we show that it is generally NP-hard to decide for a specific network whether the graph-theoretical characterization holds for all extreme points.

Keywords: Network Flow; Differential constraints; Extremal solutions; Cactus graph; Energy grids; Linearized Load Flow Model; Wheatstone bridge; 05C21; 90C35; 90B10; 05C83; 05C05 (search for similar items in EconPapers)
Date: 2025
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10957-025-02792-4 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:joptap:v:207:y:2025:i:3:d:10.1007_s10957-025-02792-4

Ordering information: This journal article can be ordered from
http://www.springer. ... cs/journal/10957/PS2

DOI: 10.1007/s10957-025-02792-4

Access Statistics for this article

Journal of Optimization Theory and Applications is currently edited by Franco Giannessi and David G. Hull

More articles in Journal of Optimization Theory and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().