Learning Geometric Combinatorial Optimization Problems using Self-attention and Domain Knowledge

TL;DR

This paper introduces a neural network model utilizing self-attention and domain knowledge to effectively solve geometric combinatorial optimization problems, demonstrating competitive results on multiple geometric tasks.

Contribution

The paper presents a novel neural network architecture with a new attention mechanism and masking scheme tailored for geometric COPs, improving learning efficiency and solution quality.

Findings

Effective in solving Delaunay triangulation, convex hull, and TSP.

Achieves competitive approximation performance.

Utilizes domain knowledge for better geometric constraint satisfaction.

Abstract

Combinatorial optimization problems (COPs) are an important research topic in various fields. In recent times, there have been many attempts to solve COPs using deep learning-based approaches. We propose a novel neural network model that solves COPs involving geometry based on self-attention and a new attention mechanism. The proposed model is designed such that the model efficiently learns point-to-point relationships in COPs involving geometry using self-attention in the encoder. We propose efficient input and output sequence ordering methods that reduce ambiguities such that the model learns the sequences more regularly and effectively. Geometric COPs involve geometric requirements that need to be satisfied. In the decoder, a new masking scheme using domain knowledge is proposed to provide a high penalty when the geometric requirement of the problem is not satisfied. The proposed neural net is a flexible framework that can be applied to various COPs involving geometry. We conduct experiments to demonstrate the effectiveness of the proposed model for three COPs involving geometry: Delaunay triangulation, convex hull, and the planar Traveling Salesman problem. Our experimental results show that the proposed model exhibits competitive performance in finding approximate solutions for solving these problems.