Learning to Solve NP-Complete Problems: A Graph Neural Network for Decision TSP

Abstract

Graph Neural Networks (GNN) are a promising technique for bridging differential programming and combinatorial domains. GNNs employ trainable modules which can be assembled in different configurations that reflect the relational structure of each problem instance. In this paper, we show that GNNs can learn to solve, with very little supervision, the decision variant of the Traveling Salesperson Problem (TSP), a highly relevant N P-Complete problem. Our model is trained to function as an effective message-passing algorithm in which edges (embedded with their weights) communicate with vertices for a number of iterations after which the model is asked to decide whether a route with cost < C exists. We show that such a network can be trained with sets of dual examples: given the optimal tour cost C * , we produce one decision instance with target cost x% smaller and one with target cost x% larger than C * . We were able to obtain 80% accuracy training with −2%, +2% deviations, and the same trained model can generalize for more relaxed deviations with increasing performance. We also show that the model is capable of generalizing for larger problem sizes. Finally, we provide a method for predicting the optimal route cost within 2% deviation from the ground truth. In summary, our work shows that Graph Neural Networks are powerful enough to solve N P-Complete problems which combine symbolic and numeric data.

Graph Colouring Meets Deep Learning: Effective Graph Neural Network Models for Combinatorial Problems

“…Because of that, we chose to model the k-colorability problem in a Graph Neural Network framework [16], a seminal formalisation which already leveraged the capability of dealing with several types of nodes. GNN models were already proven to be promising on solving relational problems, even when dealing with numeric information, such as the travelling salesperson problem (TSP) [22].…”

Graph neural network

“…分类 应用 节点分类 子图匹配、突变检测、网页排序 [10] 、团定位、二级蛋白质结构预测 [15] 网络垃圾邮件分类 [46] 、bAbI、规则发现 [16,18] 、文本挖掘 [47,48] 目标定位 [49] 、图像分类 [50] 、规则发现 [18] 、引文网络 [17,22] 、疾病预测 [51] Euclid 问题、文本分类 [21] 、知识图谱分类 [22] 、社团探测 [23] 、矩阵补全 [23,52] 组合优化 (TSP 问题、SAT 问题) [53][54][55][56] 、相近二进制码检测 [57] 、交通预测 [58] 链路预测 引文网络 [59,60] 、信息传播预测 [6] 、超图链路预测 [61] 图生成 小规模图生成 [62] 、化学分子图自动生成 [63] 社交网络、2D-网格图、蛋白质结构预测 [30] 、生成特定化学特性的分子 [28] 结构本身的拓扑信息, 影响最终的预测结果. 图神经网络则是直接处理图结构数据, 并且在迭代的过程 中始终利用了图本身的拓扑信息, 因此较之前的方法取得了更好的结果.…”

“…除此之外, 图神经网络还在其他的监督节点分类问题中被广泛地应用. 不仅给出了一些 NP-难 问题的近似算法, 如 TSP 问题 [55] 和 SAT 问题 [56] 等, 还在其他领域有许多重要的应用, 如文本挖 掘 [47,48] 、目标定位 [49] 、图像分类 [50] 、规则发现 [18] 、引文网络 [17] 和疾病预测 [51] 等. 这充分说明图神 经网络对于解决监督节点分类问题是一个行之有效的方法, 并且已经被广泛地应用在各个领域之中.…”

Fast and flexible design of novel proteins using graph neural networks

“…As there had been little previous work in using neural networks to solve CSPs (24,25), we first had to devise a network architecture that would be well-suited for this problem. In order to facilitate this search, we focused on designing a neural network capable of solving Sudoku puzzles, which is a welldefined CSP (25) for which predictions made by the network can easily be verified. We treat Sudoku puzzles as graphs having 81 nodes, corresponding to squares on the Sudoku grid, and 1701 edges, corresponding to pairs of nodes that cannot be assigned the same number (Fig.…”