Graph Learning for Combinatorial Optimization: A Survey of State-of-the-Art

Abstract

Graphs have been widely used to represent complex data in many applications, such as e-commerce, social networks, and bioinformatics. Efficient and effective analysis of graph data is important for graph-based applications. However, most graph analysis tasks are combinatorial optimization (CO) problems, which are NP-hard. Recent studies have focused a lot on the potential of using machine learning (ML) to solve graph-based CO problems. Most recent methods follow the two-stage framework. The first stage is graph representation learning, which embeds the graphs into low-dimension vectors. The second stage uses machine learning to solve the CO problems using the embeddings of the graphs learned in the first stage. The works for the first stage can be classified into two categories, graph embedding methods and end-to-end learning methods. For graph embedding methods, the learning of the the embeddings of the graphs has its own objective, which may not rely on the CO problems to be solved. The CO problems are solved by independent downstream tasks. For end-to-end learning methods, the learning of the embeddings of the graphs does not have its own objective and is an intermediate step of the learning procedure of solving the CO problems. The works for the second stage can also be classified into two categories, non-autoregressive methods and autoregressive methods. Non-autoregressive methods predict a solution for a CO problem in one shot. A non-autoregressive method predicts a matrix that denotes the probability of each node/edge being a part of a solution of the CO problem. The solution can be computed from the matrix using search heuristics such as beam search. Autoregressive methods iteratively extend a partial solution step by step. At each step, an autoregressive method predicts a node/edge conditioned to current partial solution, which is used to its extension. In this survey, we provide a thorough overview of recent studies of the graph learning-based CO methods. The survey ends with several remarks on future research directions.

A walk through of time series analysis on quantum computers

“…Note that autoregressive models can be also combined with quantum neural networks as done with the classical neural networks [32]. Also note that autoregressive models can be formulated as combinatorial optimization problems in the framework of graph learning [33]. In addition, it is shown that quadratic unconstrained optimization formulation can be used for forecasting in finance [34].…”

Optimizing Tensor Network Contraction Using Reinforcement Learning

“…Table 10 compares the total cost of finding the optimal contraction path and the actual time of executing the contraction for different algorithms and networks. Recall that for QC applications, each network has to be contracted multiple times to get a high-fidelity estimate of the circuit output [Peng et al, 2021]. Here, we assume that we contract each network = 10 6 times [Pan et al, 2021b].…”

“…Following the recent success of ML for combinatorial optimization [Dai et al, 2017, Li et al, 2018, Sato et al, 2019, Almasan et al, 2019, Nair et al, 2020, Cappart et al, 2021, Peng et al, 2021, we devise a novel and effective Reinforcement Learning (RL) approach to solve TNCO. We formulate the problem as a Markov Decision Process (MDP): The state space is defined as a space of graphs, the action space is the set of edges to be contracted, and the transition function maps a graph to a contracted graph based on the selected edge.…”

Complexity and data mining in dental research: A network medicine perspective on interceptive orthodontics

“…The procedure aims to map nodes, so the similarity in the embedding space approximates similarity in the network. 81,82 Training can be unsupervised,…”

“…It means to map nodes to a d‐dimensional embedding space (low dimensional space rather than the actual dimension of the graph) to embed close to each other similar nodes in the graph. The procedure aims to map nodes, so the similarity in the embedding space approximates similarity in the network 81,82 . Training can be unsupervised, semi‐supervised or supervised.…”