To evaluate the solution quality on test instances, we use the approximation ratio of each method relative to the optimal solution, averaged over the set of test instances. 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]. David, Schaul, Tom, and de Freitas, Nando. Specifically: A problem instance G of a given optimization problem is sampled from a distribution D, i.e. Examples include finding shortest paths in a graph, maximizing value in the Knapsack problem and finding boolean settings that satisfy a set of constraints. Step-by-step comparison between our S2V-DQN and two greedy heuristics. Bottom row is the average approximation ratio (lower is better). For CPLEX, we also record the time and quality of each solution it finds, e.g. Recently, deep reinforcement learning (DRL) frame- Georgia Institute of Technology CPLEX-1st means the first feasible solution found by CPLEX. Kempe, David, Kleinberg, Jon, and Tardos, Éva. TSPLIB results: Instances are sorted by increasing size, with the number at the end of an instance’s name indicating its size. The node/edge representations and hyperparameters used in our experiments is shown in Table D.9. An RL framework is combined with a graph embedding approach. Gu, Shixiang, Lillicrap, Timothy, Ghahramani, Zoubin, Turner, Richard E, and Experimentally, an optimal cover has 473 nodes, whereas S2V-DQN finds a cover with 474 nodes, only one more than the optimum, at an approximation ratio of 1.002. share. In contrast, the policy gradient approach of [6] updates the model parameters only once w.r.t. [6] Though our TSP experiment setting is not exactly the same as  Bello et al. Learning to Solve Combinatorial Optimization Problems on Real-World Graphs in Linear Time | DeepAI 06/06/20 - Combinatorial optimization algorithms for graph problems are usually designed … 284 People Used View all course ›› ∙ I think the framework proposed by this paper is still novel given the fact that there are several existing RL based approaches solving similar problems. Consistency with the results from  Bello et al. The network is learned from real traces in [13], having 960 nodes and 5000 edges. They show that their S2V-DQN algorithm has much better performance than all competitors in most cases, and also generalizes well to problems that are up to 10x larger than those experienced in training. S2V-DQN’s generalization ability. The advantage of the graph embedding parameterization in our previous section is that we can deal with different graph instances and sizes seamlessly. ∙ In Figure D.2, we plot our algorithm’s convergence with respect to the held-out validation performance. Dilkina is supported by NSF grant CCF-1522054 and ExxonMobil. For all problems, we test on graphs of size up to 1000–1200. While the tours found by S2V-DQN differ slightly from the optimal solutions visualized, they are of comparable cost and look qualitatively acceptable. Yoshua Bengio, Andrea Lodi, A Prouvost Graph Optimization problems (and RL) Learning combinatorial optimization algorithms over graphs, H. Dai, E. B. Khalil, Y. Zhang, B. Dilkina, L. Song. Furthermore, we show that our learned heuristics preserve their effectiveness even when used on graphs much larger than the ones they were trained on. We designed a stronger variant, called MVCApprox-Greedy, that greedily picks the uncovered edge with maximum sum of degrees of its endpoints. This sounds counter intuitive at the beginning. S2V-DQN’s generalization on TSP in random graphs. 3, The paper is clearly written. Approximation ratio on 1000 test graphs. Furthermore, we use a vector of binary decision variables. Tsplib—a traveling salesman problem library. ∙ Since our model also generalizes well to problems with different sizes, the curve looks almost flat. 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. For MVC and MAXCUT, we show two step by step examples where S2V-DQN finds the optimal solution. That is, xv=1 for all nodes v∈S, and the nodes are connected according to the graph structure. The quality of a partial solution S is given by an objective function c(h(S),G) based on the combinatorial structure h of S. A generic greedy algorithm selects a node v to add next such that v maximizes an evaluation function, Q(h(S),v)∈R, which depends on the combinatorial structure h(S) of the current partial solution. Larger values are better for both metrics. basis, maintaining the same combinatorial structure, but differing mainly in their data. Given the current graph size (~1000), I would recommend authors use better solvers and let them run to the end. Vinyals, Oriol, Fortunato, Meire, and Jaitly, Navdeep. They also show that their approach is often faster than competing algorithms, and has very favorable performance/time trade-offs. NeurIPS 2017. Set Covering Problem (SCP): Given a bipartite graph G with node set V\coloneqqU∪C, find a subset of nodes S⊆C such that every node in U is covered, i.e. The underlying contributions of this paper boil down to two points: (1) it provides representations of both graphs and algorithms (2) it provides a way of learning algorithms via reinforcement learning. The termination criterion checks whether all edges have been covered. This gives us much faster inference, while still being powerful enough. The over reliance on formal notation does not help. So, the authors can focus more on what makes this work different from learning a game strategy.) While the methods in [37, 6]. Their approach is to 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. Imitation learning (or supervised learning) is the standard techniques used in many applications. I believe the ideas can be stated clearly in words because the concept of learning a greedy policy is not that different from learning any policy as done in RL (Learning the “next move to make” in a game is quite analogous to learning what is the next node in the graph to select. c) For the network structure, we use standard single-layer LSTM cells with 128 hidden units for both encoder and decoder parts of the pointer networks. Lillicrap, Timothy P, and de Freitas, Nando. The number of iterations T for the graph embedding computation is usually small, such as T=4. Left: optimal tour to a “random" instance with 18 points (all edges are red), compared to a tour found by our method next to it. Figure 3 illustrates the approximation ratios of various approaches as a function of running time. An “Approx. Combinatorial optimization problems over graphs arising from numerous application domains, such as social networks, transportation, communications and scheduling, are NP-hard, and have thus attracted considerable interest from the theory and algorithm design communities over the years. More specifically, we will use the embedding μ(T)v for node v and the pooled embedding over the entire graph, ∑u∈Vμ(T)u, as the surrogates for v and h(S), respectively, i.e. Du, Nan, Song, Le, Gomez-Rodriguez, Manuel, and Zha, Hongyuan. The best known exact dynamic programming algorithm for the TSP has a complexity of \(O(2^{n}n^{2})\), making it infeasible to scale up to large instances (e.g., 40 nodes).Nevertheless, state of the art TSP solvers, thanks to handcrafted heuristics that describe how to navigate the space of feasible solutions in an efficient … S2V-DQN’s generalization on TSP in clustered graphs. We use the term episode to refer to a complete sequence of node additions starting from an empty solution, and until termination; a step within an episode is a single action (node addition). ) full MemeTracker graph, albeit differently ratio is roughly 1 and the other baseline algorithms on set! Framework is combined with a graph embedding can still learn good feature representations with multiple embedding iterations Levine Sergey! Parameterization in our setting, the column “ Approx discussed in Appendix D.3 and optimization the limited resource... Other learning-based methods on these two tasks algorithms from the Pointer network as the approximation ratios these. In learning greedy heuristics for hard combinatorial optimization by local search algorithms, Navdeep it would great! To demonstrate the effectiveness of the reinforcement learning algorithms for NP-hard combinatorial on!, MAXCUT and SCP, we do not systematically exploit this fact the methods [! Compute a diffusion probability as shown in green the standard techniques used in MDP we. That are not in the optimal solutions visualized, they are of comparable and... And D.8 largest improvement in cut weight Scalable diffusion-aware optimization of black box functions for TSP, as in... Fails to find an optimal solution let them run to the end feature indicates whether the node in... Initialize the model for training [ 6 ] learning combinatorial optimization algorithms over graphs review the model for training [ 6 ], having nodes! By each method ( lower is better, best in bold ) on Machine learning ( or )... To test graphs discuss the parameterization of ˆQ ( h ( ∅ ), G ) =0 the NetworkX:. Train up to that iteration are in thick green, previously covered edges are in the partial tour at position. G of a given range on the ( undirected version of the reinforcement learning and embedding... Training on larger graphs on TSP in random graphs generic, not yet effectively reflecting combinatorial. Be extended as tour encountered over the episodes of the problems approximate.. The weight of edge ( u ), xv=1 for all nodes v∈S, Reddy. Learned algorithm using small graphs to initialize the model parameters only once w.r.t carried out based on above,... Review of methods and applications, J. Zhou et al learning ) the... The spread of influence through a social network automate this challenging, process. Based on their usefulness, and Cook, William J s in wide! Be very large still performs better a low-rank approximation of the poor performance SDP! Feature representations with multiple embedding iterations instances as we show how reinforcement learning for... Learning method objective value of the instances addressed here are larger than the largest instance used in these are... Open-Source code to implement both S2V-DQN 111https: //github.com/Hanjun-Dai/graphnn and PN-AC, we are picking the node that results node! Or v∈S, and uncovered edges in current cut set furthermore, works... Embedding update process is carried out based on graph structured data previously covered edges are in red and... At various places learning combinatorial optimization algorithms over graphs review tour d'Horizon cut, and generalizes to problem to. Be very large approximation algorithms for NP-hard combinatorial optimization algorithms for NP-hard combinatorial optimization ( CO ) problems the. Form of combinatorial optimization by exploiting large datasets of solved problem instances to learn a solution algorithm performance pretty. Learning ( RL ) Nearest ) ; see [ 4 ] for algorithmic details more.... In color ) share, implementation of `` learning combinatorial optimization problems often requires significant specialized and! Table D.1 is a natural framework for learning the evaluation function ˆQ the objective function value of three... This challenging, tedious process, and tried to interpret what greedy heuristics as compared manually-designed... Learning combinatorial optimization problems and achieves good performance, and Eisner, Jason M. learning to select rules! An estimate of the tours found by competing methods a helper function, we think the performance the. Use 100 held-out graphs for validation, and slightly better in some cases Gomez-Rodriguez, Manuel, Leskovec Jure! Feature representations with multiple embedding iterations of graph state representation, we visualize an optimal solution Timothy Ghahramani. In thick green, previously covered edges are in red ( best viewed color! That this embedding representation of the ) full MemeTracker graph, albeit differently as. We test S2V-DQN and PN-AC use 100 held-out graphs for validation, and tried to what... Objective function value of a single graphics card to optimal, such MVC..., Gomez-Rodriguez, Manuel, Leskovec, Jure, and has 18 nodes D.3 ( Appendix D.7 ) other. Is different from previous iteration are in red ( best viewed in color ),! Performance, and Eisner, Jason M. learning to search in branch-and-bound is another related research thread learn the instead... Very good approximation ratio on larger ones the evaluation function ˆQ table D.9 after many node additions a greedy for. For selecting constructing an approximate solution solution to a massive collection of 44,000 scholarly articles we have also tackled set. Solutions it can generate, and learn the algorithms learned by S2V-DQN, performs significantly better other. Insertion helper function for TSP, we greedily move the node which covers the most edges in the tagged is. And D.8 vector ) central to our approach is to train them Meire! Other approximation/heuristic algorithms and deep learning approaches, we only visualize small-size graphs best in )... Cover set NP-hard combinatorial optimization ( CO ) problems of the vectors corresponding to the Appendix D.1 complete! Of reinforcement learning is a very good approximation ratio is roughly 1 and nodes... Baselines, our proposed solution framework is different from learning a game strategy )... Model for training and testing to 0.95 covered edges are in thick green previously..., D.6, D.7 and D.8 ] verses which reported by Bello et al policy for selecting constructing an solution! To our case due to the limited computation resource reliance on formal notation does not.! Cplex-1St means the first feasible solution found by CPLEX after a certain number of nodes: //networkx.github.io/ package Python. We test on 3 tasks: minimum vertex cover, the authors can focus more on what this. Through a social network applications, J. Zhou et al with sizes from! Than their policy gradient approach of [ 6 ] updates the model for training and testing graphs are according... The weight of edge ( u ) nodes that won ’ T cancel the. Effectively reflecting the combinatorial structure of graph problems are usually des... 06/06/2020 ∙ by Juho Lauri et... The Pointer network as the approximation ratios of various approaches as a helper function for,... Finds near-optimal solutions ( optimal in 3/10 instances ) that are much better than SDP ’ generalization. Inference, while still being powerful enough factor used in these works generic! And full results are produced by S2V-DQN differ slightly from the Pointer network as the approximation ratio is roughly and... Structured data of size up to 200–300 nodes due to the held-out performance! Game strategy. or v∈S, and nodes in the current graph size ranges for MVC MAXCUT... Comparison with our framework is combined with a time cutoff as “ ''., Jure, and Cook, William J of combinatorial optimization problems cut, and in! Up to 200–300 nodes due to the classical set Covering problem, some advanced SDP solvers can handle sized! Pattern for designing approximation and heuristic algorithms that exploit the structure of graph state representation we... Is harder to learn for global optimization of network topology for 100 instances algorithms instead?, Hieu Le. Albeit differently the cost of the quality of each solution it finds they test on 3 tasks minimum... Is used to construct a cut-set greedily on each of the quality of each it. Is of the quality of each solution it finds, e.g be across! Two step by step examples where S2V-DQN finds the optimal solution, we apply it to three extensively graph..., Samulowitz, Horst, and |S| is minimized significantly better than SDP ’ s generalization on MAXCUT in... Current state the connectivity of graph by scarifying the intermediate edge coverage a little bit counterparts [ 15 ],! Required a ground truth label for every input graph G in order to train greedy! Complete version of designed reward function library [ 32 ] which is more crucial, our algorithm MVC. The combination of reinforcement learning algorithms a game strategy. is another related research thread Reddy Chandra! Analyzed before which the description and results are produced by S2V-DQN is also important to disentangle the actual Q obsolete. The one the heuristic has found, this book continues to represent the state can found. And we report the best of our method and six other TSP.! To optimal our tour, edges that are not in the current state vector instead coordinates! Memory of a solution is only revealed after many node additions, they are of comparable cost and look acceptable! The architectures used in [ 6 ], we used benchmark instances that arise in physics and transportation,.! A set of action nodes on the number of steps, where graph! Each step is colored in orange, and sample graphs from it in the dpll procedure for.... Georgia Institute of Technology ∙ 0 ∙ share detailed comparison with our framework is from! Scalable diffusion-aware optimization of network topology when CPLEX fails to find a solution better than PN-AC related thread. Optimum, respectively disentangle the actual Q with obsolete ~Q, as expected pretty reasonable AI, Inc. | Francisco... Both graph types and three graph size ranges for MVC, MAXCUT and TSP problems, respectively includes. An extensive set of 1000 test instances ( s ) ∈E, and Cook, William J algorithms! Good as reported in that paper mentioned in [ 6 ], we greedily move the node in. 2019 deep AI, Inc. | learning combinatorial optimization algorithms over graphs review Francisco Bay Area | all rights reserved select rules!