Mining maximal frequent itemsets is one of the most fundamental problems in data mining. In this paper we study the complexity-theoretic aspects of maximal frequent itemset mining, from the perspective of counting the number of solutions. We present the first formal proof that the problem of counting the number of distinct maximal frequent itemsets in a database of transactions, given an arbitrary support threshold, is #P-complete, thereby providing strong theoretical evidence that the problem of mining maximal frequent itemsets is NP-hard. This result is of particular interest since the associated decision problem of checking the existence of a maximal frequent itemset is in P.We also extend our complexity analysis to other similar data mining problems dealing with complex data structures, such as sequences, trees, and graphs, which have attracted intensive research interests in recent years. Normally, in these problems a partial order among frequent patterns can be defined in such a way as to preserve the downward closure property, with maximal frequent patterns being those without any successor with respect to this partial order. We investigate several variants of these mining problems in which the patterns of interest are subsequences, subtrees, or subgraphs, and show that the associated problems of counting the number of maximal frequent patterns are all either #P-complete or #P-hard.

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Abstract. Although RDF/XML has been widely recognized as the standard vehicle for representing semantic information on the Web, an enormous amount of semantic data is still being encoded in HTML documents that are designed primarily for human consumption and not directly amenable to machine processing. This paper seeks to bridge this semantic gap by addressing the fundamental problem of automatically annotating HTML documents with semantic labels. Exploiting a key observation that semantically related items exhibit consistency in presentation style as well as spatial locality in template-based content-rich HTML documents, we have developed a novel framework for automatically partitioning such documents into semantic structures. Our framework tightly couples structural analysis of documents with semantic analysis incorporating domain ontologies and lexical databases such as WordNet. We present experimental evidence of the effectiveness of our techniques on a large collection of HTML documents from various news portals.