Complexity of Source-Sink Monotone 2-Parameter Min Cut

TL;DR

This paper investigates the complexity of parametric min cuts in source-sink monotone networks, revealing that with multiple parameters, the number of distinct min cuts can grow exponentially, contrasting with the single-parameter case.

Contribution

It demonstrates that the number of distinct min cuts can be exponential with two parameters, resolving an open question in the field.

Findings

Number of min cuts can be exponential with two parameters.

Single-parameter min cuts are linearly bounded.

Nested min cuts property holds for multiple parameters.

Abstract

There are many applications of max flow with capacities that depend on one or more parameters. Many of these applications fall into the "Source-Sink Monotone" framework, a special case of Topkis's monotonic optimization framework, which implies that the parametric min cuts are nested. When there is a single parameter, this property implies that the number of distinct min cuts is linear in the number of nodes, which is quite useful for constructing algorithms to identify all possible min cuts. When there are multiple Source-Sink Monotone parameters and the vector of parameters are ordered in the usual vector sense, the resulting min cuts are still nested. However, the number of distinct min cuts was an open question. We show that even with only two parameters, the number of distinct min cuts can be exponential in the number of nodes.