# Solving hard cut problems via flow-augmentation

## Abstract

*flow augmentation*. Our technique is applicable to problems that can be phrased as a search for an (edge) (

*s, t*)-cut of cardinality at most

*k*in an undirected graph

*G*with designated terminals

*s*and

*t*.

*Z*⊆

*E*(

*G*) of size at most

*k*such that

*G*–

*Z, s*and

*t*are in distinct connected components,

*Z*connects two distinct connected components of

*G*–

*Z*, and

*Z*⊆

_{s, t}*Z*as those edges

*e ∊ Z*for which there exists an (

*s, t*)-path

*P*with

_{e}*E*(

*P*) ∩

_{e}*Z*= {

*e*}, then

*Z*separates

_{s, t}*s*from

*t*.

*k*

^{(1)}(|

*V*(

*G*)| + |

*E*(

*G*)|) add a number of edges to the graph so that with probably at least 2

^{–(k log k)}no added edge connects two components of

*G*–

*Z*, and

*Z*becomes a

_{s, t}*minimum cut*between

*s*and

*t*.

*s, t*)-cut of cardinality at most

*k*and of minimum possible weight (assuming edge weights in

*G*). While the problem is NP-hard in general, it easily reduces to the maximum flow / minimum cut problem if we additionally assume that

*k*is the minimum possible cardinality of an (

*s, t*)-cut in

*G*. Hence, we immediately obtain that the aforementioned problem admits an 2

^{(k log k)}

*n*

^{(1)}-time randomized fixed-parameter algorithm.

*Coupled Min-Cut*. This problem emerges out of the study of FPT algorithms for Min CSP problems (see below), and was unamenable to other techniques for parameterized algorithms in graph cut problems, such as Randomized Contractions, Treewidth Reduction or Shadow Removal.

*i*-Chain SAT problem. We are able to show that every problem Min SAT(Γ) is either (1) FPT, (2) W[1]-hard, or (3) able to express the soft constraint (

*u*→

*v*), and thereby also the min-cut problem in directed graphs. All the W[1]-hard cases were known or immediate, and the main new result is an FPT algorithm for a generalization of Coupled Min-Cut. In other words, flow-augmentation is powerful enough to let us solve every fixed-parameter tractable problem in the class, except those that explicitly encompass directed graph cuts.

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**ISBN (Online)**: 978-1-61197-646-5

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#### History

**Published online**: 7 January 2021

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