Adaptive Network Flow with $k$-Arc Destruction

TL;DR

This paper demonstrates that allowing adaptive flow reorientation transforms the maximum residual flow problem with $k$-arc destruction from NP-hard to polynomial time for any fixed $k$, providing a significant complexity reduction.

Contribution

It introduces the concept that adaptivity in flow reorientation makes the problem tractable for fixed $k$, contrasting with the non-adaptive case which is NP-hard.

Findings

Adaptive flow reorientation reduces complexity to polynomial time for fixed $k$.

Maximum residual flow with $k$-arc destruction is NP-hard without adaptivity.

Theoretical proof of polynomial solvability under adaptive conditions.

Abstract

When a flow is not allowed to be reoriented the Maximum Residual Flow Problem with $k$-Arc Destruction is known to be $NP$-hard for $k=2$. We show that when a flow is allowed to be adaptive the problem becomes polynomial for every fixed $k$.