Abstract
The unicost set covering problem and the attribute reduction problem are NP-complete problems. In this paper, the relationship between these two problems are discussed. Based on the transformability between attribute reductions and minimal solutions in unicost set covering models, two methods are provided. One is to induce an information table from a given unicost set covering model. With no doubt, it shows that the unicost set covering problem can be investigated by rough set theory. The other is to induce a unicost set covering model from a given information table. Similarly, it shows that the attribute reduction problem can be studied by set covering theory. As an application of the proposed theoretical results, a rough set heuristic algorithm is presented for the unicost set covering problem.
Highlights
We briefly introduce the contents of the following sections
By constructing an induced information table of a given USCP, we find that computing a minimal solution or a minimum solution in a USCP can be converted to computing an attribute reduction or a minimum attribute reduction in the constructed information table. erefore, the USCP can be converted into the APR in rough set theory
By constructing an induced unicost set covering (USC) model according to an information table, we find that calculating an attribute reduction or a minimum attribute reduction of an information table is equal to calculating a minimal solution or a minimum solution of the constructed USCP
Summary
Some basic concepts of SCP, USCP, information table, and ARP in rough sets are reviewed. A mininal solution of the USCP is an optimum sensors setting scheme for Example 1. Attribute reduction is one of the core problems in rough sets. For B ⊆ A, if B ∩ d(xi, xj) ≠ ∅, it means that the attribute set B can discern the object pair (xi, xj) [18]. RED(IT) denotes the set of all attribute reductions of an information table IT. Am) is an attribute reduction of IT, and all the prime implicants of discernibility function fS(a1, a2, . We can use the disjunction ∨ and conjunction ∧ operations to compute all the attribute reductions of an information table. One can obtain all the attribute reductions of an information table by computing the discernibility function using Boolean operations.
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