âLearning combinatorial optimization algorithms over graphs,â in Proceedings of the 31st International Conference on Neural Information Processing Systems, NIPS'17, (Long Beach, CA), 6351â6361. Selecting action v corresponds to adding a node of G to the current partial solution, which results in collecting a reward r(S,v). For MVC, MAXCUT and SCP, we represent nodes based on the adjacency matrix of the graph. In the end, if one terminates after T iterations, each node embedding μ(T)v will contain information about its T-hop neighborhood as determined by graph topology, the involved node features and the propagation function F. An illustration of two iterations of graph embedding can be found in Figure 1. Empirically, inserting a node u in the partial tour at the position which increases the tour length the least is a better choice. We show that the agent is not only picking the node with large degree, but also trying to maintain the connectivity after removal of the covered edges. Furthermore, we show that our learned heuristics preserve their effectiveness even when used on graphs much larger than the ones they were trained on. The fitted Q-iteration approach has been shown to result in faster learning convergence when using a neural network as a function approximator [33, 28], a property that also applies in our setting, as we use the embedding defined in Section 3.2. In contrast, the policy gradient approach of [6] updates the model parameters only once w.r.t. We set the rank as 8, so that each node in the input sequence is represented by a 8-dimensional vector. Ioannis, Wierstra, Daan, and Riedmiller, Martin A. 0 Bibliographic details on Learning Combinatorial Optimization Algorithms over Graphs. For all problems, we test on graphs of size up to 1000–1200. Mnih, Volodymyr, Kavukcuoglu, Koray, Silver, David, Rusu, Andrei A, Veness, Consistency with the results from Bello et al. Since many combinatorial optimization problems, such as the set covering problem, can be explicitly or implicitly formulated on graphs, we believe that our work opens up a new avenue for graph algorithm design and discovery with deep learning. All three paradigms seldom exploit a common trait of real-world optimization problems: Neural combinatorial optimization with reinforcement learning. In comparison, our work promotes an even tighter integration of learning and optimization. For the Minimum Vertex Cover (MVC) problem, we generate random Erdős-Renyi (edge probability 0.15) and Barabasi-Albert (average degree 4) graphs of various sizes, and use the integer programming solver CPLEX 12.6.1 with a time cutoff of 1 hour to compute optimal solutions for the generated instances. Bottom row is the average approximation ratio (lower is better). g) For the baseline function in the actor-critic algorithm, we tried the critic network in our implementation, but it hurts the performance according to our experiments. combinatorial optimization, A General Framework for Charger Scheduling Optimization Problems, Learning Heuristics over Large Graphs via Deep Reinforcement Learning, A Generic Bet-and-run Strategy for Speeding Up Traveling Salesperson and ∙ 3, The paper is clearly written. We let Q∗ denote the optimal Q-function for each RL problem. Note that the “optimal" value used in the computation of approximation ratios may not be truly optimal (due to the solver time cutoff at 1 hour), and so CPLEX’s solutions do typically get worse as problem size grows. S2V-DQN’s generalization on MVC problem in BA graphs. Challenge consists of 10 specific data anlaysis tasks. ∙ Experimental analysis of heuristics for the stsp. Alternatively, a package delivery company routes trucks on a daily basis in a given city; thousands of similar optimizations need to be solved, since the underlying demand locations can differ. Boyan and Moore [7] use regression to learn good restart rules for local search algorithms. The approximation ratio of a solution S to a problem instance G is defined as With the development of machine learning in various fields, it can also be applied to combinatorial optimization problems, automatically discovering generic and fast heuristic algorithms based on training data, and requires fewer theoretical and empirical knowledge. Implementation of Learning Combinatorial Optimization Algorithms over Graphs, by Hanjun Dai et al. We implemented PN-AC to the best of our capabilities. Combinatorial optimization problems over graphs arising from numerous application domains, such as social networks, transportation, telecommunications and scheduling, are NP-hard, and have thus attracted considerable interest from the theory and algorithm design communities over the years. Graves, Alex, Wayne, Greg, Reynolds, Malcolm, Harley, Tim, Danihelka, Ivo, We modify existing open-source code to implement both S2V-DQN 111https://github.com/Hanjun-Dai/graphnn and PN-AC 222https://github.com/devsisters/pointer-network-tensorflow. That is, xv=1 for all nodes v∈S, and the nodes are connected according to the graph structure. Imitation learning (or supervised learning) is the standard techniques used in many applications. S2V-DQN’s generalization on TSP in clustered graphs. 0 Then, the partial solution S will be extended as. In fact, of Karp’s 21 problems in the seminal paper on reducibility [19], 10 are decision versions of graph optimization problems, while most of the other 11 problems, such as set covering, can be naturally formulated on graphs. Applegate, David, Bixby, Robert, Chvatal, Vasek, and Cook, William. Our code is publicly available 333https://github.com/Hanjun-Dai/graph_comb_opt. Ratio of Best Solution" value of 1.x% means that the solution found by CPLEX if given the same time as a certain heuristic (in the corresponding row) is x% worse, on average. The SCP is also related to the diffusion optimization problem on graphs; for instance, the proof of hardness in the classical [20] paper uses SCP for the reduction. For the Maximum Cut (MAXCUT) problem, we use the same graph generation process as in MVC, and augment each edge with a weight drawn uniformly at random from [0,1]. Our graph embedding parameterization ˆQ(h(S),v;Θ) from Section 3 will then be a function approximation model for it, which will be learned via n-step Q-learning. Furthermore, existing works typically use the policy gradient for training [6], a method that is not particularly sample-efficient. For example, a state is defined as a sequence of action nodes on the graph. There, the output of the embedding is linked with a softmax-layer, so that the parameters can by trained end-to-end by minimizing the cross-entropy loss. The framework is set up in such a way that the policy will aim to optimize the objective function of the original problem instance directly. R(ˆS)=∑|ˆS|i=1r(Si,vi) is equal to c(h(ˆS),G); Policy: based on ˆQ, a deterministic greedy policy π(v|S):=\argmaxv′∈¯¯¯SˆQ(h(S),v′) will be used. Furthermore, with limited ablation analysis, we don't have a good understanding on what's going on. After a few steps of recursion, the network will produce a new embedding for each node, taking into account both graph characteristics and long-range interactions between these node features. a) For the input data, we use mini-batches of 128 sequences with 0-paddings to the maximal input length (which is the maximal number of nodes) in the training data. Scalable influence estimation in continuous-time diffusion networks. Join one of the world's largest A.I. Here, both the state of the graph and the context of a node v can be very complex, hard to describe in closed form, and may depend on complicated statistics such as global/local degree distribution, triangle counts, distance to tagged nodes, etc. To handle different graph sizes, we use a singular value decomposition (SVD) to obtain a rank-8 approximation for the adjacency matrix, and use the low-rank embeddings as inputs to the pointer network. SCP is interesting because it is not a graph problem, but can be formulated as one. Table D.1 is a complete version of Table 2 that appears in the main text. Then, for a given method M that terminates in T seconds on a graph G and returns a solution with approximation ratio R, we asked the following 2 questions: If CPLEX is given the same amount of time T for G, how well can CPLEX do? Learning to learn for global optimization of black box functions. For the MVC and MAXCUT problems, we generate Erdős-Renyi (ER) [11] and Barabasi-Albert (BA) [1] graphs which have been used to model many real-world networks. We will use fitted Q-learning to learn a greedy policy that is parametrized by the graph embedding network. When we consider only those graphs for which CPLEX could find a better solution, S2V-DQN’s solutions take significantly more time for CPLEX to beat, as compared to MaxcutApprox and SDP. general methodology applied to resource-constrained scheduling. For instance, on graphs with 1200 nodes, we can find the solution of MVC within 11 seconds using a single GPU, while getting an approximation ratio of 1.0062. They test on 3 tasks: minimum vertex cover, maximum cut, and traveling salesman problem. S2V-DQN achieves an average approximation ratio of 1.001, only slightly behind LP, which achieves 1.0009, and well ahead of Greedy at 1.03. Instead of performing a gradient step in the loss of the current sample as in (6, ), stochastic gradient descent updates are performed on a random sample of tuples drawn from. We can see our algorithm will also favor the large degree nodes, but it will also do something smartly: instead of breaking the graph into several disjoint components, our algorithm will try the best to keep the graph connected. All approximation ratios reported in the paper are with respect to the best (possibly optimal) solution found by the solvers within 1 hour. That is, in many applications, values of the coefficients in the objective function or constraints can be thought of as being sampled from the same underlying distribution. We have also tackled the Set Covering Problem, for which the description and results are deferred to Appendix B. ∙ Table 2 summarizes the results, and full results are in Appendix D.3. Machine learning for branch-and-bound. combinatorial optimization with reinforcement learning and neural networks. Also in the intermediate steps, the agent seldom chooses a node which would cancel out the edges that are already in the cut set. An estimate of the quality of the solution resulting from adding a node to partial solution. Algorithm design pattern. Combinatorial Optimization with Graph Convolutional Networks and Guided Tree Search Zhuwen Li Intel Labs Qifeng Chen ... Learning-based approaches have the potential to yield more effective empirical algorithms for NP-hard problems by learning from large datasets. Google Scholar How long does CPLEX need to find a solution of same or better quality than the one the heuristic has found? The advantage of the graph embedding parameterization in our previous section is that we can deal with different graph instances and sizes seamlessly. share, In this paper, we propose a deep reinforcement learning framework called... Lillicrap, Timothy P, and de Freitas, Nando. ∙ Also for TSP with insertion helper function, we find it works better with negative version of designed reward function. The authors aim to solve problems in combinatorial optimization by exploiting large datasets of solved problem instances to learn a solution algorithm. For MVC and MAXCUT, the graph is of the ER type and has 18 nodes. For experiments of PN-AC across all tasks, we follow the configurations provided in [6]: It would be great to have a title which is more specific. We show our framework can be applied to a diverse range of (u,s)∈E, and |S| is minimized. 3, It would be great to show how the propagation step in the struct2vec model affects the performance on graphs which is not fully connected. As such, the 1-step update may be too myopic. Kempe, David, Kleinberg, Jon, and Tardos, Éva. We further examined the algorithms learned by S2V-DQN, and tried to interpret what greedy heuristics have been learned. For instance, S2V-DQN discovers an algorithm for MVC where nodes are selected to balance between their degrees and the connectivity of the remaining graph (Appendix Figures D.4 and D.7). All algorithms report a single solution at termination, whereas CPLEX reports multiple improving solutions, for which we recorded the corresponding running time and approximation ratio. For the first question, the column “Approx. 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