Equivalent spectral theory for fundamental graph cut problems

TL;DR

This paper develops a unified spectral graph theory framework for fundamental graph cut problems, linking eigenpairs to optimal cuts through a novel set-pair Lovász extension and Dinkelbach scheme.

Contribution

It introduces an equivalent eigenproblem formulation for various graph cut problems using set-pair Lovász extension and establishes a unified nonlinear spectral theory framework.

Findings

Eigenpairs correspond to optimal graph cuts.

Unified spectral framework for multiple cut problems.

Analysis of eigenvector structure and spectral properties.

Abstract

We introduce and develop equivalent spectral graph theory for several fundamental graph cut problems including maxcut, mincut, Cheeger cut, anti-Cheeger cut, dual Cheeger problem and their useful variants. A specified strategy for achieving an equivalent eigenproblem is proposed for a general graph cut problem via the set-pair Lov\'asz extension and the Dinkelbach scheme. For a class of 2-cut and 3-cut problems, we reveal the intrinsic difference-of-submodularity for the fractional formulations and show that their set-pair Lov\'asz extensions yield equivalent difference-of-convex structures. Building on the Dinkelbach scheme, we finally establish a unified research roadmap for nonlinear spectral theory that provides a one-to-one correspondence between certain eigenpairs and the optimal graph cut problems. The finer structure of the eigenvectors, the Courant nodal domain theorem and the graphic feature of eigenvalues are studied systematically in the setting of these new nonlinear eigenproblems.