Unsupervised Ordering for Maximum Clique

TL;DR

This work addresses the maximum clique problem by replacing binary membership labels with a learning-based vertex ordering learned in a permutation framework. It encodes MCP as a continuous geometric optimization using $M(A)=J-I-A$ and a distance-based weight $\mathbf{D}_{\text{Clique}}$, optimizing $\mathcal{L}_{\text{Clique}} = \langle \mathbf{T}^T (J-I-A) \mathbf{T}, \mathbf{D}_{\text{Clique}} \rangle$ with a soft permutation $\mathbb{T}$ and then converting to a hard permutation $\mathbf{P}$ for search via the Gumbel-Sinkhorn/Hungarian pipeline. The learned clique-oriented ordering is integrated into the MaxCliqueDyn branch-and-bound solver to replace the traditional degree-based ordering, improving pruning and reducing search steps, especially on denser graphs, while maintaining competitive wall-clock times. The approach demonstrates generalization to different graph sizes through zero-padding and training on larger instances, with inference overhead diminishing as problem size grows, indicating practical viability for enhancing exact solvers.

Abstract

We propose an unsupervised approach for learning vertex orderings for the maximum clique problem by framing it within a permutation-based framework. We transform the combinatorial constraints into geometric relationships such that the ordering of vertices aligns with the clique structures. By integrating this clique-oriented ordering into branch-and-bound search, we improve search efficiency and reduce the number of computational steps. Our results demonstrate how unsupervised learning of vertex ordering can enhance search efficiency across diverse graph instances. We further study the generalization across different sizes.

Figures (5)

• Figure 1: Graph representations and their corresponding matrices. (a) The original graph, (b) the corresponding adjacency matrix $A$, (c) $\mathbf{M}(A) = J - I - A$ (where $J$ is the all-ones matrix and $I$ is the identity matrix); (d) graph (a) with reordered nodes, (e) the corresponding adjacency matrix $A'$, (f) $\mathbf{M}(A') = J - I - A'$.
• Figure 2: Overview of the unsupervised learning framework for TSP. The model takes graph features as input and processes them through a GNN. The objective is formulated within a permutation framework. The output provides a heat map that guides the subsequent search.
• Figure 3: (a): Visualization of a 6x6 chessboard with a king positioned at A6; (b) the Chebyshev distance matrix $C_6$, where each element represents the minimum number of moves required for a king to travel between corresponding squares; (c) $\overline{C_6} = 5 - C_6$, where the elements at top left have larger weights. $C_n[i,j] = \text{max}\{i,j\}$ - 1 and $\overline{C_n}[i,j] = n - 1- C_n = n - \text{max}\{i,j\}$
• Figure 4: Adjacency matrix visualization of the graph: (a) random ordering, (b) clique-oriented ordering, and (c) matrix sorted by non-increasing degree.
• Figure 5: Adjacency matrix of the first 50 nodes of the graph: (a) random ordering, (b) clique-oriented ordering, and (c) matrix sorted by non-increasing degree.

Theorems & Definitions (2)

• Lemma 1
• proof