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Robust Weighted Partial Maximum Satisfiability Problem: Challenge to \(\varSigma _{2}^{P}\)-Complete Problem

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PRICAI 2022: Trends in Artificial Intelligence (PRICAI 2022)

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Abstract

This paper introduces a new problem called the Robust Maximum Satisfiability problem (R-MaxSAT), as well as its extension called the Robust weighted Partial MaxSAT (R-PMaxSAT). In R-MaxSAT (or R-PMaxSAT), a problem solver called defender hopes to maximize the number of satisfied clauses (or the sum of their weights) as the standard MaxSAT/partial MaxSAT problem, although she must ensure that the obtained solution is robust (In this paper, we use the pronoun “she” for the defender and “he” for the attacker). We assume an adversary called the attacker will flip some variables after the defender selects a solution. R-PMaxSAT can formalize the robust Clique Partitioning Problem (robust CPP), where CPP has many real-life applications. We first demonstrate that the decision version of R-MaxSAT is \(\varSigma _2^P\)-complete. Then, we develop two algorithms to solve R-PMaxSAT, by utilizing a state-of-the-art SAT solver or a Quantified Boolean Formula (QBF) solver as a subroutine. Our experimental results show that we can obtain optimal solutions within a reasonable amount of time for randomly generated R-MaxSAT instances with 30 variables and 150 clauses (within 40 s) and R-PMaxSAT instances based on CPP benchmark problems with 60 vertices (within 500 s).

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Notes

  1. 1.

    These variables can overlap with the defender’s decision/auxiliary variables.

  2. 2.

    A similar problem is considered in [26], while they assume a coalition’s value is given as a black-box function called a characteristic function.

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Acknowledgements

This work is supported by JSPS KAKENHI Grant Numbers JP19H04175, JP20H00609, and JP22K19813.

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Authors and Affiliations

  1. Kyushu University, Fukuoka, Japan

    Tomoya Sugahara, Kaito Yamashita, Miyuki Koshimura & Makoto Yokoo

  2. EPF, Graduate School of Engineering, Cachan, France

    Nathanaël Barrot

Authors
  1. Tomoya Sugahara
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  2. Kaito Yamashita
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  3. Nathanaël Barrot
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  4. Miyuki Koshimura
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  5. Makoto Yokoo
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Correspondence to Tomoya Sugahara .

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Editors and Affiliations

  1. CSIRO Australian e-Health Research Centre, Brisbane, QLD, Australia

    Sankalp Khanna

  2. Shanghai Jiao Tong University, Shanghai, China

    Jian Cao

  3. University of Tasmania, Hobart, TAS, Australia

    Quan Bai

  4. University of Technology Sydney, Sydney, NSW, Australia

    Guandong Xu

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Sugahara, T., Yamashita, K., Barrot, N., Koshimura, M., Yokoo, M. (2022). Robust Weighted Partial Maximum Satisfiability Problem: Challenge to \(\varSigma _{2}^{P}\)-Complete Problem. In: Khanna, S., Cao, J., Bai, Q., Xu, G. (eds) PRICAI 2022: Trends in Artificial Intelligence. PRICAI 2022. Lecture Notes in Computer Science, vol 13629. Springer, Cham. https://doi.org/10.1007/978-3-031-20862-1_2

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  • DOI: https://doi.org/10.1007/978-3-031-20862-1_2

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