# Most vital segment barriers

## TL;DR

This paper extends the concept of vital network segments to geometric environments, analyzing the complexity and providing algorithms for placing barriers that significantly impact flows or paths within polygonal domains.

## Contribution

It introduces a geometric analogue of vital arcs, establishes hardness results, and develops efficient algorithms for barrier placement problems.

## Key findings

- Identifies computational hardness in certain barrier placement scenarios.
- Provides polynomial-time algorithms for specific cases.
- Clarifies the boundary between hard and tractable problems.

## Abstract

We study continuous analogues of "vitality" for discrete network flows/paths, and consider problems related to placing segment barriers that have highest impact on a flow/path in a polygonal domain. This extends the graph-theoretic notion of "most vital arcs" for flows/paths to geometric environments. We give hardness results and efficient algorithms for various versions of the problem, (almost) completely separating hard and polynomially-solvable cases.