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Finding Optimal Exact Reducts
Hassan AbouEisha
Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and
Technology, KAUST, Thuwal 23955-6900, Saudi Arabia
Keywords:
Exact Reducts, Covering, Dynamic Programming, Knowledge Reduction, Feature Selection.
Abstract:
The problem of attribute reduction is an important problem related to feature selection and knowledge dis-
covery. The problem of finding reducts with minimum cardinality is NP-hard. This paper suggests a new
algorithm for finding exact reducts with minimum cardinality. This algorithm transforms the initial table to a
decision table of a special kind, apply a set of simplification steps to this table, and use a dynamic program-
ming algorithm to finish the construction of an optimal reduct. I present results of computer experiments for a
collection of decision tables from UCI ML Repository. For many of the experimented tables, the simplification
steps solved the problem.
1 INTRODUCTION
The problem of constructing a reduct with minimum
cardinality for a given decision table is one of the
key problems in the rough set theory (Pawlak, 1992;
Pawlak and Skowron, 2007; Skowron and Rauszer,
1992). It is also related to knowledge discovery, fea-
ture selection and data mining. It is well known
that this problem is NP-hard (Skowron and Rauszer,
1992). Different approximate approaches for find-
ing minimal reducts have been presented in literature
(Hoa and Son, 1996; Wroblewski, 1995). However,
based on results of Feige for the set cover problem
(Feige, 1998), it is possible to show that, under some
natural assumptions about the class NP, the approxi-
mation ratio of the best approximate polynomial time
algorithm for reduct optimization is near to the nat-
ural logarithm on the number of pairs of rows (ob-
jects) with different decisions in the decision table
(Moshkov et al., 2008). Therefore, the improvement
of exact algorithms for reduct optimization continues
to be an important issue.
In this paper, I propose an algorithm which trans-
forms the initial decision table T into a new table T
(1)
with the same set of reducts. After that, the algo-
rithm simplifies the table T
(1)
by removal of some
rows and columns (there are some analogies between
this part of the algorithm and attribute reduction al-
gorithms using discernibility matrix (Skowron and
Rauszer, 1992)). As a result, we have a new deci-
sion table T
(2)
and a subset A of the set of removed
attributes. The union of A and an arbitrary reduct
with minimum cardinality for T
(2)
is a reduct with
minimum cardinality for T. The problem of reduct
optimization for T
(2)
is equivalent to the problem of
minimization of decision tree depth for T
(2)
. The last
problem can be resolved by a dynamic programming
algorithm (Alkhalid et al., 2011).
A similar approach but with essentially more com-
plicated simplification part (transformation of T
1
into
T
2
) was introduced in (AbouEisha et al., 2013). This
approach has more reduction rules that are continu-
ously executed until none of them can be applied.
In this paper, I consider a number of decision ta-
bles from UCI ML Repository (A. Asuncion, 2007)
and construct, for these tables, reducts with minimum
cardinality using the proposed algorithm.
The paper consists of four sections. In Section
2, the algorithm for reduct optimization is described.
Section 3 contains results of computer experiments,
and Section 4 provides a short conclusion.
2 MINIMIZATION OF REDUCT
CARDINALITY
In this section, I consider basic notions and describe
the algorithm for reduct optimization.
A decision table T is a rectangular table with n
columns labeled with conditional attributes f
1
, . . . , f
n
.
Rows of this table are filled by nonnegative inte-
gers which are interpreted as values of conditional at-
149
AbouEisha H..
Finding Optimal Exact Reducts.
DOI: 10.5220/0005035501490153
In Proceedings of the International Conference on Knowledge Discovery and Information Retrieval (KDIR-2014), pages 149-153
ISBN: 978-989-758-048-2
Copyright
c
2014 SCITEPRESS (Science and Technology Publications, Lda.)