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Computing all minimal hitting sets by subset recombination

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Abstract

In model-based diagnosis from first principles, the efficient computation of all minimal hitting sets (MHS) as candidates for the conflict component sets of a device is a vital task. However, deriving all MHS is NP-hard. In this paper, the principle of “Divide and Conquer” is used to decompose a large family of conflict sets into many smaller sub-families. To efficiently merge the sub-MHS to give sub-families of conflict sets, the relations between the sub-MHS and sub-families of conflict sets are exploited. Based on this, a new method called Subset-Rec-MHS is proposed. In theory, our method based on sub-MHS recombination generally has lower complexity than that based on whole MHS families, as it avoids a large number of set unions and comparisons (to minimize the family of hitting sets). Compared with the direct merge of whole MHS families, the proposed approach reduces the computation time by a factor of approximately \(\frac {7}{16}\). Experimental results on both synthetic examples and ISCAS-85 benchmark circuit conflict sets show that, in many cases, our approach offers better performance than previous algorithms.

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Notes

  1. In this paper, to discriminate different types of symbols, we use lower-caseletters (such as e), upper-case letters (such as S), and calligraphic letters (such as\(\mathcal {F}\))to denote the basic elements of a set, basic sets including some basic elements, and families of some basic sets, respectively.

  2. Generally, in model-based diagnosis, there is an assumption that the connections between components are workingnormally.

  3. As we use a two-way divide and merge strategy to derive the MHS, we assume that the family of conflict sets is organized as 2X groups, with each group having elements that are different from those in other groups.

  4. The rationale for using such synthetic instances is twofold. First, the cardinality of each MHS for such instances is generally not very large, and so generating such MHS is usually fast. Second, these structures are properly regular and can easily be extended to larger sizes.

  5. The package of conflict sets for ISCAS-85 benchmark circuits can be downloaded from http://www.fe.up.pt/~rma/benchmarks2.zip or extracted from the Lydia software from https://general-diagnostics.com/downloads.php. The package can also be requested by email from the authors.

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Acknowledgements

The authors would like to acknowledge the anonymous referees for their constructive comments, which considerably improved the quality of the manuscript.

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Authors and Affiliations

  1. College of Mathematics, Physics, and Information Engineering, Zhejiang Normal University, Jinhua, China

    Xiangfu Zhao

  2. Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University, Changchun, China

    Dantong Ouyang & Liming Zhang

Authors
  1. Xiangfu Zhao
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  2. Dantong Ouyang
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  3. Liming Zhang
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Correspondence to Xiangfu Zhao.

Additional information

This work was supported in part by Zhejiang Provincial Natural Science Foundation of China under Grant No. LY16F020004 and the National Natural Science Foundation of China under Grant Nos. 61003101, 61272208, and 61472369.

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Zhao, X., Ouyang, D. & Zhang, L. Computing all minimal hitting sets by subset recombination. Appl Intell 48, 257–270 (2018). https://doi.org/10.1007/s10489-017-0971-7

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