Computing metric hulls in graphs

We prove that, given a closure function the smallest preimage of a closed set can be calculated in polynomial time in the number of closed sets. This confirms a conjecture of Albenque and Knauer and implies that there is a polynomial time algorithm to compute the convex hull-number of a graph, when all its convex subgraphs are given as input. We then show that computing if the smallest preimage of a closed set is logarithmic in the size of the ground set is LOGSNP-complete if only the ground set is given. A special instance of this problem is computing the dimension of a poset given its linear extension graph, that was conjectured to be in P. The intent to show that the latter problem is LOGSNP-complete leads to several interesting questions and to the definition of the isometric hull, i.e., a smallest isometric subgraph containing a given set of vertices $S$. While for $|S|=2$ an isometric hull is just a shortest path, we show that computing the isometric hull of a set of vertices is NP-complete even if $|S|=3$. Finally, we consider the problem of computing the isometric hull-number of a graph and show that computing it is $\Sigma^P_2$ complete.Comment: 13 pages, 3 figure

CICLAD: A Fast and Memory-efficient Closed Itemset Miner for Streams

Mining association rules from data streams is a challenging task due to the (typically) limited resources available vs. the large size of the result. Frequent closed itemsets (FCI) enable an efficient first step, yet current FCI stream miners are not optimal on resource consumption, e.g. they store a large number of extra itemsets at an additional cost. In a search for a better storage-efficiency trade-off, we designed Ciclad,an intersection-based sliding-window FCI miner. Leveraging in-depth insights into FCI evolution, it combines minimal storage with quick access. Experimental results indicate Ciclad's memory imprint is much lower and its performances globally better than competitor methods.Comment: KDD2

International Workshop "What can FCA do for Artificial Intelligence?" (FCA4AI at IJCAI 2013, Beijing, China, August 4 2013)

International audienceThis second edition of the FCA4AI workshop (the first edition was associated to the ECAI 2012 Conference, see http://www.fca4ai.hse.ru/), shows again that there are many AI researchers interested in FCA. Formal Concept Analysis (FCA) is a mathematically well-founded theory aimed at data analysis and classification. FCA allows one to build a concept lattice and a system of dependencies (implications) which can be used for many AI needs, e.g. knowledge processing involving learning, knowledge discovery, knowledge representation and reasoning, ontology engineering, as well as information retrieval and text processing. Thus, there exist many natural links between FCA and AI. Accordingly, the focus in this workshop was on how can FCA support AI activities (knowledge processing) and how can FCA be extended in order to help AI researchers to solve new and complex problems in their domains

On Computing the Minimal Generator Family for Concept Lattices and Icebergs

Abstract. Minimal generators (or mingen) constitute a remarkable part of the closure space landscape since they are the antipodes of the closures, i.e., minimal sets in the underlying equivalence relation over the powerset of the ground set. As such, they appear in both theoretical and practical problem settings related to closures that stem from fields as diverging as graph theory, database design and data mining. In FCA, though, they have been almost ignored, a fact that has motivated our long-term study of the underlying structures under different perspectives. This paper is a two-fold contribution to the study of mingen families associated to a context or, equivalently, a closure space. On the one hand, it sheds light on the evolution of the family upon increases in the context attribute set (e.g., for purposes of interactive data exploration). On the other hand, it proposes a novel method for computing the mingen family that, although based on incremental lattice construction, is intended to be run in a batch mode. Theoretical and empirical evidence witnessing the potential of our approach is provided.