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Equivalent spectral theory for fundamental graph cut problems

Sihong Shao, Chuan Yang, Dong Zhang, Weixi Zhang

TL;DR

This work develops a unified nonlinear spectral framework for fundamental graph cuts, including Cheeger, maxcut, dual Cheeger, and anti-Cheeger problems. By combining the set-pair Lovász extension with the Dinkelbach scheme, it converts fractional combinatorial objectives into equivalent continuous DC programs and derives nonlinear eigenproblems whose eigenpairs coincide with optimal cuts in key settings. It provides two complementary Cheeger formulations via the graph 1-Laplacian and a signless variant, extends higher-order dual Cheeger theory, and establishes maxcut and anti-Cheeger eigenproblems with nodal-domain type results, all framed within a cohesive roadmap for nonlinear spectral graph theory. The results connect submodularity, convex analysis, and variational principles to yield tractable spectral characterizations and principled procedures for approximating NP-hard graph-cut problems with potential practical impact in clustering and partitioning tasks.

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 Lovász 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 Lovász 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.

Equivalent spectral theory for fundamental graph cut problems

TL;DR

This work develops a unified nonlinear spectral framework for fundamental graph cuts, including Cheeger, maxcut, dual Cheeger, and anti-Cheeger problems. By combining the set-pair Lovász extension with the Dinkelbach scheme, it converts fractional combinatorial objectives into equivalent continuous DC programs and derives nonlinear eigenproblems whose eigenpairs coincide with optimal cuts in key settings. It provides two complementary Cheeger formulations via the graph 1-Laplacian and a signless variant, extends higher-order dual Cheeger theory, and establishes maxcut and anti-Cheeger eigenproblems with nodal-domain type results, all framed within a cohesive roadmap for nonlinear spectral graph theory. The results connect submodularity, convex analysis, and variational principles to yield tractable spectral characterizations and principled procedures for approximating NP-hard graph-cut problems with potential practical impact in clustering and partitioning tasks.

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 Lovász 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 Lovász 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.

Paper Structure

This paper contains 12 sections, 48 theorems, 152 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notations
  3. A new perspective of nonlinear eigenproblems
  4. Equivalent continuous optimization for graph cut
  5. Equivalent 2-step iterative scheme
  6. Equivalent nonlinear eigenproblem
  7. Equivalent eigenproblems for Cheeger cut
  8. Graph 1-Laplacian: the first equivalent eigenproblem for Cheeger cut
  9. A new equivalent eigenproblem for Cheeger cut
  10. Higher order spectral theory for dual Cheeger problems
  11. Spectral theory for maxcut
  12. An equivalent eigenproblem for anti-Cheeger cut

Key Result

Theorem 1

Every set-pair function $f: \mathcal{P}_2(V)\rightarrow[0,+\infty)$ is the difference of two strictly submodular functions, and its corresponding set-pair Lovász extension $f^L$ is a DC (difference-of-convex) function.

Figures (2)

  • Figure 1: A bipartition $(S,S^c)$ of the vertex set $V$ is displayed in red bullets ($S$) and blue circles ($S^c$). For the Cheeger cut or the anti-Cheeger cut, the remaining uncut edges (in black) are divided equally in $S$ and $S^c$ (balanced and judicious!). For the mincut or the maxcut, the remaining uncut edges are all contained in one part (biased!).
  • Figure 2: A research roadmap for the construction of an equivalent continuous spectral theory for the graph cut problem.

Theorems & Definitions (91)

  • Theorem 1
  • Theorem 2
  • Definition 1: set-pair Lovász extension CSZZ-Lovasz18
  • Proposition 1: equivalent continuous formulation for 2-cut CSZZ-Lovasz18BuhlerRangapuramSetzerHein2013
  • Proposition 2: equivalent continuous formulation for 3-cut CSZZ-Lovasz18
  • Definition 2: set-pair function and strictly submodularity CSZZ-Lovasz18
  • Proposition 3: set-pair Lovász extension and convexity CSZZ-Lovasz18
  • Proposition 4: SY24
  • Proposition 5: CSZZ-Lovasz18
  • Definition 3
  • ...and 81 more