[Paper Review] Dual Cheeger Constants, Signless 1-Laplacians and Maxcut

Sihong Shao, Chuan Yang+1 more|arXiv (Cornell University)|Jul 2, 2016

Graph theory and applications20 references8 citations

TL;DR

This paper introduces the signless 1-Laplacian on graphs and establishes its nonlinear spectral theory, proving that the dual Cheeger constant $ h^+ $ equals $ 1 - \mu_1^+ $, where $ \mu_1^+ $ is the first eigenvalue of the signless 1-Laplacian. The work extends the Courant nodal domain theorem and develops a set-pair Lovász extension to bridge discrete optimization and continuous function minimization, enabling an efficient inverse power method for the recursive spectral cut algorithm in 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 $0.769$.

Motivation & Objective

• To develop a nonlinear spectral theory for the signless 1-Laplacian on graphs, analogous to the classical 1-Laplacian theory.
• To establish a precise relationship between the dual Cheeger constant $ h^+ $ and the first eigenvalue $ \mu_1^+ $ of the signless 1-Laplacian.
• To extend the Courant nodal domain theorem to the signless 1-Laplacian with a modified nodal domain definition.
• To construct a set-pair version of the Lovász extension that captures the dual Cheeger problem’s objective function.
• To enable an efficient inverse power method for computing $ h^+ $, supporting a practical recursive spectral cut algorithm for the maxcut problem.

Proposed method

• Define the signless 1-Laplacian functional $ I^+(\mathbf{x}) = \sum_{u \sim v} |x_u + x_v| $ on graph functions with unit $ \ell^1 $-norm.
• Prove that $ \mu_1^+ = \inf_{\mathbf{x} \in X} I^+(\mathbf{x}) $, where $ X $ is the unit $ \ell^1 $-sphere, and show $ \mu_1^+ = 0 $ if and only if the graph is bipartite.
• Establish a set-pair Lovász extension that maps discrete set-pair functions to continuous functions, enabling the recovery of $ h^+ = 1 - \mu_1^+ $.
• Use subdifferential analysis to show local linearity of $ I^+(\mathbf{x}) $, which enables the inverse power method for efficient computation of $ \mu_1^+ $.
• Modify the classical Courant nodal domain theorem to apply to the signless 1-Laplacian, defining nodal domains based on sign changes of eigenvectors.
• Leverage the co-area formula and graph transformation techniques to derive lower bounds on $ I^+(\mathbf{x}) $, leading to $ \inf I^+(\mathbf{x}) \geq 1 - h^+(G) $.

Experimental results

Research questions

• RQ1 Can the dual Cheeger constant $ h^+ $ be exactly characterized via a nonlinear spectral theory analogous to the 1-Laplacian?
• RQ2 Is there a spectral functional whose first eigenvalue satisfies $ \mu_1^+ = 1 - h^+ $?
• RQ3 How can the Lovász extension be generalized to handle set-pair functions arising in the dual Cheeger problem?
• RQ4 Can the local linearity of the signless 1-Laplacian functional enable a practical inverse power method for computing $ h^+ $?
• RQ5 Does the Courant nodal domain theorem extend to the signless 1-Laplacian with a modified nodal domain definition?

Key findings

• The dual Cheeger constant $ h^+ $ satisfies $ h^+ = 1 - \mu_1^+ $, where $ \mu_1^+ $ is the first eigenvalue of the signless 1-Laplacian, turning the dual Cheeger inequality into an equality.
• The first eigenvalue $ \mu_1^+ = 0 $ if and only if the graph is bipartite, providing a spectral characterization of bipartiteness.
• The Courant nodal domain theorem is extended to the signless 1-Laplacian, with nodal domains defined by the sign structure of the eigenvector.
• A set-pair version of the Lovász extension is constructed, which allows the exact recovery of the relationship $ h^+ = 1 - \mu_1^+ $.
• The functional $ I^+(\mathbf{x}) $ is locally linear in any direction, enabling the inverse power method for efficient computation of $ \mu_1^+ $ and thus $ h^+ $.
• The inverse power method provides an efficient, non-SDP implementation of the recursive spectral cut algorithm for the maxcut problem with guaranteed cost > 1/2.