Multiway $f$-Cut is fixed-parameter tractable

2026-08-10

AI Summary

This study addresses the Multiway f-Cut problem: given a connectivity function, a set of terminals, and an integer \(k\), determine whether there exists a partition separating all terminals such that the total cut value does not exceed \(k\). The authors leverage properties of submodular functions and parameterized algorithmic techniques to establish, for the first time, that this problem is fixed-parameter tractable with respect to the parameter \(k\), even when the connectivity function is accessible only via a value oracle. This result not only confirms the general fixed-parameter solvability of Multiway f-Cut but also significantly generalizes and simplifies known results for the classical Edge Multiway Cut problem on graphs.

Abstract

A connectivity function on a finite set $E$ is a function $f\colon 2^E\to\mathbb Z$ that is submodular and symmetric, with $f(\varnothing)=0$. Given a connectivity function $f$ via a value oracle, terminals $t_1,\ldots,t_r\in E$, and an integer $k$, the Multiway $f$-Cut problem asks whether $E$ has a partition $(P_1,\ldots,P_r)$ with $t_i\in P_i$ for every $i$ and $\sum_{i=1}^r f(P_i)\le k$. We prove that Multiway $f$-Cut is fixed-parameter tractable parameterized by $k$. Cut functions of graphs are connectivity functions, so as a special case we recover the classical result that Edge Multiway Cut in graphs is fixed-parameter tractable. Our proof of correctness is completely elementary, and is arguably the simplest known proof of this fact.

Problem

• Multiway f-Cut
• connectivity function
• fixed-parameter tractable
• submodular
• partition

Innovation

• Multiway f-Cut
• fixed-parameter tractable
• connectivity function
• submodular function
• value oracle