# Multi-Objective Optimization for High-Dimensional Maximal Frequent Itemset Mining
## 1 Introduction
[23] designed an artificial cell division algorithm, which was very successful in solving multi-channel spatial searching tasks and was superior to the Apriori algorithm with respect to small-scaled transaction sets. [PERSON] et al. [24] applied the PSO algorithm to explore a frequent itemset from a transaction set named FoodMart2000. This FoodMart2000 transaction set contained 12,100 transactions and had 34 dimensions (number of items). [PERSON] et al. [25] explored frequent itemset in nursery transaction sets by using the quantum group evolutionary algorithm. The number of nursery transactions and dimensions were 12,960 and 32, respectively. [PERSON] et al. [26] strengthened the random searching performance of the PSO algorithm and explored the frequent itemset in a transaction set, which has 1000 transactions and 5 dimensions. None of the aforementioned algorithms have been applied to high-dimensional and mass transaction sets. To test the feasibility of the algorithms, accident and marketing transaction sets of real business supermarkets were selected as the testing objects of the high-dimensional mining of frequent itemsets in this paper. The number of transactions and dimensions in accident transaction sets were 340,183, and 468. The transaction length of the accident transaction set had uniform distributions accompanied by a high quantity of transactions and dimensions. This accident transaction set was as representative and could be used as the test object. The transaction set of the business supermarket contained 65,435 marketing records, which were equal to 18,548 transaction sets. The dimension of commodities was even as high as 5547. Although Weka [27] was the most famous software tool of ML and DM tasks, its algorithm can either be directly applied to a dataset through its interface or used in ones' own Java codes. However, it is inapplicable to big and high-dimensional data, and the operation time cannot wait. KEEL [28] also provided many existing algorithms and testing data that are only applicable to data-type transaction sets and are thus inapplicable to the frequent item mining of Boolean-type transaction sets. In order to mine all frequent itemsets, maximal frequent itemsets are important. If the maximum frequent itemsets are used as the search targets of the algorithm, the algorithm can find as many frequent itemsets as possible with fewer targets. This is because maximal frequent itemsets contain frequent itemsets at their maximum capacity, and all subsets of maximal frequent itemsets are still frequent itemsets. The definition of maximal frequent itemsets is given in Section 2. In addition, relaxed functional dependencies (RFDs) are properties expressing important relationships among data [29]. Thanks to the introduction of approximations in data comparison and/or validity, they can capture constraints useful for several purposes, such as the identification of data inconsistencies or patterns of semantically related data. Nevertheless, RFDs can only provide benefits if they can be automatically discovered from data. [PERSON] [30] presented an RFD discovery algorithm relying on a lattice structured search space, previously used for FD discovery, new pruning strategies, and a new candidate RFD validation method. An experimental evaluation demonstrates the discovery performances of the proposed algorithm on real datasets, also providing a comparison with other algorithms. The remainder of this paper is as follows. Section 2 introduces the related concepts, including association rules, frequent itemsets and maximal frequent itemsets, the mathematical description of the problem and so on. Section 3 states the operating framework of the proposed algorithm and the details of each part. In Section 4, two experiments are used to illustrate the effectiveness of the algorithm. Section 5 discusses the advantages and disadvantages of the algorithm and gives the conclusion.