Deciding Monotone Duality and Identifying Frequent Itemsets in Quadratic Logspace

TL;DR

This paper demonstrates that the monotone duality problem and related hypergraph duality testing can be solved within quadratic logspace, providing new complexity bounds for problems in databases and data mining.

Contribution

It introduces a quadratic logspace algorithm for monotone duality testing, improving understanding of the problem's computational complexity.

Findings

Duality testing for hypergraphs is feasible in DSPACE[log^2 n].

The results imply efficient decision procedures for frequent itemset problems.

New complexity bounds are established for related database and data mining problems.

Abstract

The monotone duality problem is defined as follows: Given two monotone formulas f and g in iredundant DNF, decide whether f and g are dual. This problem is the same as duality testing for hypergraphs, that is, checking whether a hypergraph H consists of precisely all minimal transversals of a simple hypergraph G. By exploiting a recent problem-decomposition method by Boros and Makino (ICALP 2009), we show that duality testing for hypergraphs, and thus for monotone DNFs, is feasible in DSPACE[log^2 n], i.e., in quadratic logspace. As the monotone duality problem is equivalent to a number of problems in the areas of databases, data mining, and knowledge discovery, the results presented here yield new complexity results for those problems, too. For example, it follows from our results that whenever for a Boolean-valued relation (whose attributes represent items), a number of maximal frequent itemsets and a number of minimal infrequent itemsets are known, then it can be decided in quadratic logspace whether there exist additional frequent or infrequent itemsets.