Dual Cheeger Constants, Signless 1-Laplacians and Maxcut

TL;DR

The paper develops a spectral framework for dual Cheeger constants $h^+(G)$ and $\widehat{h}^+(G)$ via signless 1-Laplacians $\Delta_1^+$ and $\widehat{\Delta}_1^+$, linking these constants to eigenvalues $\mu_1^+$ and $\widehat{\mu}_1^+$. It introduces inverse power methods (IP and $\widehat{IP}$) that converge to ternary eigenvectors, enabling rounding-free approximations of the dual Cheeger problems. By embedding IP/\widehat{IP} into the Recursive Spectral Cut (RSC) framework, the authors obtain higher-quality maxcut solutions on the G-set and prove a non-improvable lower bound of $0.769$ for the worst-case ratio. The work strengthens the discrete-to-continuous spectral connections in combinatorial optimization and yields practical, scalable algorithms for NP-hard maxcut problems.

Abstract

The first nontrivial lower bound of the worst-case approximation ratio for the maxcut problem was achieved via the dual Cheeger problem, whose optimal value is referred to the dual Cheeger constant $h^+$, and later improved through its modification $\widehat{h}^+$. However, the dual Cheeger problem and its modification themselves are relatively unexplored, especially lack of effective approximate algorithms. To this end, we first derive equivalent spectral formulations of $h^+$ and $\widehat{h}^+$ within the framework of the nonlinear spectral theory of signless 1-Laplacian, present their interactions with the Laplacian matrix and 1-Laplacian, and then use them to develop an inverse power algorithm that leverages the local linearity of the objective functions involved. We prove that the inverse power algorithm monotonically converges to a ternary-valued eigenvector, and provide the approximate values of $h^+$ and $\widehat{h}^+$ on G-set for the first time. The recursive spectral cut algorithm for the maxcut problem can be enhanced by integrating into the inverse power algorithms, leading to significantly improved approximate values on G-set. Finally, we show that the lower bound of the worst-case approximation ratio for the maxcut problem within the recursive spectral cut framework can not be improved beyond $0.769$.

Figures (4)

• Figure 1: The Cheeger problem given in Equation \ref{['eq:Cheeger0']} or \ref{['eq:Cheeger1']} and its relations with the $\Delta_1$- and $\Delta_2$-eigenproblems shown in Equations \ref{['egL2']} and \ref{['eq:L1']}, respectively.
• Figure 2: Operators, constants and connections investigated in this paper. The explicit reference to the graph G is neglected here for simplicity.
• Figure 3: The complex $X$ of the path graph with two vertices consists of four 0-cells (the four vertices of the square) and four 1-cells (the four sides of the square), while the corresponding refined complex $X^+$ consists of six 0-cells (black dots) and four small 1-cells (dashed lines) and two big 1-cells (dotted lines).
• Figure 4: Flowchart of the $t$-th RSC iteration for maxcut. Here $\hbox{\boldmath$x$}^{[t]}$ is the maximal eigenvector of the graph Laplacian $\Delta_2(G_t)$ on the residual graph $G_t=(V_t, E_t)$, $\hbox{\boldmath$y$}^{[t]}\in \{-1,0,1\}^{|V_t|}$ gives a ternary vector, and $\widehat{r}$ is the recursive stopping indicator defined in Equation \ref{['ratio:modified_dual_cheeger']}. The original RSC adopts the rounding procedure, 2TSCTrevisan2012 or $\widehat{\textbf{2TSC}}$Soto2015, which accepts $\hbox{\boldmath$x$}^{[t]}$ as input and outputs $\hbox{\boldmath$y$}^{[t]}$. In order to develop an enhanced RSC, this work replaces 2TSC (resp. $\widehat{\textbf{2TSC}}$) with IP (resp. $\widehat{\rm \bf{IP}}$). When RSC stops at the $N$-iteration, a series of partitions, $(L_1,R_1)$, $\ldots$, $(L_N,R_N)$, are obtained and can be used to form an approximate solution for maxcut.

Theorems & Definitions (28)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1: (Normalized) Ternary Vector
• Lemma 2.2
• proof
• proof : Proof of Theorem \ref{['th:1-eigen']}
• Remark 2.3
• Proposition 2.4
• proof
• proof : Proof of Theorem \ref{['th:1-ieq']}