# Multiway k-Cut in Static and Dynamic Graphs: A Maximum Entropy Principle Approach

## TL;DR

This paper introduces a maximum entropy principle-based algorithm for solving the multiway k-cut problem in both static and dynamic graphs, aiming to find optimal partitions while managing computational complexity and dynamic influences.

## Contribution

It proposes a novel maximum entropy relaxation approach for multiway k-cut problems, including dynamic graph control, with convergence guarantees and practical algorithms.

## Key findings

- Algorithm converges to a local minimum in static graphs.
- Effective control laws for dynamic graphs maintain low multiway k-cut values.
- Simulations demonstrate the approach's efficacy in real-world scenarios.

## Abstract

This work presents a maximum entropy principle based algorithm for solving minimum multiway $k$-cut problem defined over static and dynamic {\em digraphs}. A multiway $k$-cut problem requires partitioning the set of nodes in a graph into $k$ subsets, such that each subset contains one prespecified node, and the corresponding total cut weight is minimized. These problems arise in many applications and are computationally complex (NP-hard). In the static setting this article presents an approach that uses a relaxed multiway $k$-cut cost function; we show that the resulting algorithm converges to a local minimum. This iterative algorithm is designed to avoid poor local minima with its run-time complexity as $\sim O(kIN^3)$, where $N$ is the number of vertices and $I$ is the number of iterations. In the dynamic setting, the edge-weight matrix has an associated dynamics with some of the edges in the graph capable of being influenced by an external input. The objective is to design the dynamics of the controllable edges so that multiway $k$-cut value remains small (or decreases) as the graph evolves under the dynamics. Also it is required to determine the time-varying partition that defines the minimum multiway $k$-cut value. Our approach is to choose a relaxation of multiway $k$-cut value, derived using maximum entropy principle, and treat it as a control Lyapunov function to design control laws that affect the weight dynamics. Simulations on practical examples of interactive foreground-background segmentation, minimum multiway $k$-cut optimization for non-planar graphs and dynamically evolving graphs that demonstrate the efficacy of the algorithm, are presented.