Short communicationA memetic algorithm based on edge-state learning for max-cut
Introduction
Combinatorial optimization is a kind of mathematical optimization, it is aimed to find an optimal solution of a problem within its finite and discrete solution space subject to a given objective function. Some of the combinatorial optimization problems, such as most of the NP-hard problems, are well-known to be extremely difficult to solve. Therefore, researchers often resort to heuristic algorithms to find best possible solutions of them with reasonable resources. As one of the most classic NP-hard problems (Karp, 1972), max-cut involves dividing the vertices of a graph into two disjoint subsets to maximize the weighted sum of the edges crossing the two subsets. Solutions to this problem and its close variations (e.g., min-cut, min–max-cut, -max-cut, etc.) can be used in data mining (Ding et al., 2001), cloud computing (Bansal et al., 2014), VLSI layout (Cho et al., 1998), wireless communication-frequency scheduling (Eisenblätter, 2002), football team scheduling (Mitchell, 2003), and statistical physics (Liers et al., 2004). On the other hand, methods to solve such classic and extremely challenging problems can inspire researchers on many other similar or related problems. The problem can be formulated as follows:
Given an undirected graph (, ), and . is the set of vertices, is the set of edges. Each edge connects two different vertices and , and is associated with a weight . The problem is to divide the vertex set into two disjoint subsets and , in order to maximize the sum of the weights of edges connecting vertices belonging to different subsets. The objective function of the problem is as follows:
A solution to a max-cut problem can be represented as a vector (, , …..., ), ( ) represents the position of the th vertex, i.e., which subset ( or ) it belongs to. Due to the NP-Complete nature of max-cut (Karp, 1972), exact algorithms are only applicable to some small or special structured instances so far (Della Croce et al., 2007, Kneis and Rossmanith, 2005). As for large-scale and general instances, it is intractable to solve them exactly up to now. Therefore, researchers usually resort to approximation algorithms and heuristic algorithms. The approximation algorithms use relaxation methods (Goemans and Williamson, 1995, Krishnan and Mitchell, 2006) to provide good approximate solutions to max-cut. They often can guarantee the lower-bound of their algorithms, but they usually cannot apply to large scale instances efficiently (Burer et al., 2002). Although heuristic algorithms often are inefficient for special cases such like graphs mainly consists of cycles or Hamilton paths, and cannot guarantee the lower-bound, they can provide high quality approximate solutions with reasonable time for large-scale general cases. The heuristic algorithms can be further divided into two classes: the specialized heuristic algorithms and the metaheuristic algorithms. The specialized heuristic algorithms are customized for the max-cut problem, such as the rank-two relaxation heuristic (Burer et al., 2002) and the global equilibrium search algorithms (Shylo et al., 2015, Shylo et al., 2012). The metaheuristic algorithms first employ powerful general-purpose metaheuristic frameworks, then design the elements of the metaheuristic based on their insights into the max-cut problem. These elements include neighborhoods (moves), perturbation method, evolutionary method, and so on. They also often integrate several metaheuristics together to better utilize their distinct features regarding intensification and diversification. The metaheuristic algorithms include the local search algorithms based on Tabu search (Arráiz and Olivo, 2009, Kochenberger et al., 2013), Variable Neighborhood search (Festa et al., 2002), and Simulated Annealing (Arráiz & Olivo, 2009); the evolutionary or memetic algorithms based on scatter search (Martí et al., 2009), GRASP (Festa et al., 2002, Wu et al., 2015), and path relinking (Festa et al., 2002, Wang et al., 2012); iterated local search based algorithms (Benlic and Hao, 2013, Ma and Hao, 2017, Ma et al., 2017); etc. Recently, a two-phase metaheuristic framework has become a research focus. It includes an intensification phase and a diversification phase. It uses sophisticate local search metaheuristic such as Tabu search or (and) Variable Neighborhood search to find good local-optimum solution within their intensification phase, and uses metaheuristics such as iterated local search, grasp, or path-relinking to construct diversified and promising initial solutions for the intensification phase during the diversification phase. The whole process alternates between these two phases repeatedly. Many of such algorithms have shown great effectiveness (Ma and Hao, 2017, Ma et al., 2017, Wu et al., 2015) since they can utilize the complementary search capacity of the two phases properly. Based on its prominent feature, we also adopt this iterative two-phase framework in our metaheuristic-based algorithm. Although previous two-phase metaheuristic algorithms have provided state-of-the-art results on max-cut, none of them has employed any learning-based methods in their diversification phase, which have already shown great effectiveness on many other optimization problems regarding providing best solution within reasonable time (Fu and Hao, 2017, Khalil et al., 2017, Zhou et al., 2018, Zhou and Hao, 2017). This might be caused by the difficulties in designing effective learning operators for the max-cut problem: previous learning methods usually utilize the statistics on element positions, but as illustrated by the following example, one cannot use such statistics of max-cut solutions reasonably and easily.
Learning based methods such as Estimation of Distribution Algorithms (EDAs) (Larrañaga & Lozano, 2001) and Opposite Based Learning (OBL) (Tizhoosh, 2005) have been a hot research area of evolutionary computation since the beginning of this century. Generally, they use the statistic information of a population or previous searches to construct promising new solutions or guide future searches. There are two key issues involved in designing effective learning operators: what to learn and how to learn. What to learn prescribes what statistic information one should extract from a population; how to learn determines how to construct or partly reconstruct a solution according to these statistic information. A solution to a problem is often represented as a vector, and learning operators usually make use of the statistic information on each component of the vector independently (Fu and Hao, 2017, Zhou et al., 2018). Therefore, which kind of elements one used to represent a solution can well-define what statistic information to learn. As described above, the max-cut problem has two kinds of elements: vertices and edges. Previous state-of-the-art algorithms use the positions of vertices to represent a solution and whereupon measure the similarity between two solutions for crossover (Wang et al., 2012, Wu et al., 2015). We think it may be not an effective choice in some cases. Consider the example in Fig. 1, the vertices of solutions and are divided into two sets by two dividing lines respectively. Both solutions and are cutting their edges , , , , and (the bolded edges in Fig. 1). They are the same with regard to cutting edges, which are the only things calculated by the objective function (formula (1)). However, they are quite different according to their vertex positions. They cannot be encoded into one representation according to their vertex positions. Specifically, solution can be encoded as 01010101 or 10101010 depending on which vertex set we choose as . Similarly, solution can be encoded as 10100101 or 01011010. Solution and are quite different no matter which encoding results one chooses. This situation is caused by the symmetry of local configurations. If one rotates the local configuration on subgraph 1 in solution around the dividing line to its mirror position, then all the vertex positions of solution and will be the same. One can observe that vertex positions are not a good criterion for measuring the similarity between two solutions in this case, and sometimes they cannot provide critical information on what a solution really is. In many real-world applications, graphs consist of high-coherent, low-coupling subgraphs. In these cases, solutions can suffer similar local-configuration symmetry issues, and the statistics on vertex positions over a population will not make any sense. This difficulty may be the reason why learning operators are not employed in any previous state-of-the-art algorithms.
To deal with this issue, we use the edge-state based encoding method for learning. It encodes a solution by its edge states rather than vertex positions. A solution hence can be represented in another form: (, , …..., ), ( ) represents the state of its th edge: if it is a cutting edge, it is set to 1; otherwise 0. By this means, solution and solution are both encoded into the same representation: 11101011. This suggests that edge-states may be a more proper criterion to measure the similarity between two solutions, and they can provide more meaningful information about a solution than vertex positions do. Therefore, the statistic information on edge states may be more relevant than that on vertex positions for such structural cases.
Although the edge-state encoding is more meaningful on such structural instances, it also has posed a challenge to designing effective learning operators. Unlike vertex positions, one cannot set the state of an edge without influencing the states of some other edges in most occasions. For example, in Fig. 1, if one switches the state of from its current cutting state to non-cutting state, it will also change the states of edges {} or {, }. Due to this dependency issue, one cannot set the states of edges one by one independently just like some other learning operators (Fu and Hao, 2017, Zhou et al., 2018) perform on their solution components. To overcome this difficulty, we propose a method named post-flip edge-state learning (PF-ESL). Rather than setting edge states directly, it samples the flips on vertices. The flip on a vertex is sampled according to its capacity in increasing the similarity on edge states between the solution and a population. By iteratively sampling such flips, the solution will become more and more similar to the population with regard to edge states.
The main contributions of this paper are as follows. Firstly, previous state-of-the-art evolutionary algorithms for max-cut focus on vertex positions in their evolutionary operation, this paper proposes a new and more reasonable perspective suggesting that edge-states are the critical information of divided graphs rather than vertex positions, and introduces a novel method to measure and utilize their similarities based on it. Such a perspective is fundamental to learning based algorithms design for max-cut and other graph partitioning problems, and can shed lights on future researches. Furthermore, since max-cut is one of the most classic and fundamental NP hard problems, many real-world problems involve dividing graph data into different parts to optimize certain functions, this new perspective may inspire related or similar problems. Secondly, besides the original edge-states based perspective, and the post-flip edge-states learning (PFESL) operator based on it, our memetic algorithm also incorporates a novel evolutionary framework which alternates between EDA based Iterated Tabu search (ITS) and path relinking based genetic algorithm. Finally, the proposed algorithm provides competitive results on two mostly used benchmark sets and improves the best-known results of 6 most challenging instances.
The rest of our paper is organized as follows. Section 2 presents a detailed description on the proposed algorithm. Section 3 provides the experimental results of our algorithm comparing with other state-of-the-art algorithms, and analyzes the key features of our algorithm. Finally, Section 4 concludes the paper.
Section snippets
Solution method
This section describes our solution in detail. Section 2.1 first provides the main framework of our solution method; the other subsection then describes its components in detail. We also list all the variables used in the solution method and their descriptions as follows for reference:
P: Sections 2.1–2.2; the population.
p: Sections 2.1–2.2; the size of the population.
: Section 2.1; a constant value prescribes the possibility of using Path_relinking to construct a new solution from the
Experiments and analysis
To assess the proposed PF-ESL based memetic algorithm, we implemented it in Visual C++ language and performed the following experiments. The computer we used to run the experiments is a PC with an AMD FX 8350 CPU (4 GHZ), 4 GB RAM. The operation system is Windows 7, 32 bit. The important parameter setting of our algorithm for the two test sets are listed in Table 1.
The first experiment is aimed to evaluate the “discovery capability” (Grosso et al., 2007) of our algorithm, i.e., the ability to
Conclusions
This paper has proposed an original learning method named post-flip edge-state learning (PF-ESL) for the max-cut problem. Different from previous state-of-the-art works, it regards edge states rather than vertex positions as the key information of a solution, and calculates the statistics on them over a population for its learning operations. The idea is based on the observations that edge states and edge weights are the only factors involved in the objective function, and edge states are
Declaration of Competing Interest
One or more of the authors of this paper have disclosed potential or pertinent conflicts of interest, which may include receipt of payment, either direct or indirect, institutional support, or association with an entity in the biomedical field which may be perceived to have potential conflict of interest with this work. For full disclosure statements refer to https://doi.org/10.1016/j.eswa.2022.118077. Zhizhong Zeng reports financial support was provided by National Natural Science Foundation
Acknowledgments
This work is supported by National Natural Science Foundation of China (Grant No. 61877026); National Natural Science Foundation of China for Outstanding Youth (Grant No. 72122006); Self-determined Research Funds of CCNU from the colleges’ basic research and operation of MOE (Grant No. CCNU19ZN001); Natural Science Basic Research Program of Shaanxi (Grant No. 2020JM-089). We would give our special thanks to the anonymous referees for their professional comments and valuable suggestions that
References (33)
- et al.
Breakout local search for the max-cutproblem
Engineering Applications of Artificial Intelligence
(2013) - et al.
An exact algorithm for MAX-CUT in sparse graphs
Operations Research Letters
(2007) - et al.
Knowledge-guided local search for the prize-collecting Steiner tree problem in graphs
Knowledge-Based Systems
(2017) - et al.
An effective iterated tabu search for the maximum bisection problem
Computers & Operations Research
(2017) - et al.
Path relinking for unconstrained binary quadratic programming
European Journal of Operational Research
(2012) - et al.
A tabu search based hybrid evolutionary algorithm for the max-cut problem
Applied Soft Computing
(2015) - et al.
Improving probability learning based local search for graph coloring
Applied Soft Computing
(2018) - et al.
Frequency-driven tabu search for the maximum s-plex problem
Computers & Operations Research
(2017) - et al.
Competitive simulated annealing and tabu search algorithms for the max-cut problem
- et al.
Min-max graph partitioning and small set expansion
SIAM Journal on Computing
(2014)