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# Measure and Conquer for Max Hamming Distance XSAT

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LIPIcs.ISAAC.2019.15.pdf
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## Acknowledgements

The authors would like to thank the anonymous referees of ISAAC 2019 for useful suggestions. Furthermore, the authors would like to thank internet companies for putting services like Wolfram Alpha Equation Solver, Firefox Scratchpad and Google Scholar for free onto the internet.

## Cite As

Gordon Hoi and Frank Stephan. Measure and Conquer for Max Hamming Distance XSAT. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 15:1-15:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ISAAC.2019.15

## Abstract

XSAT is defined as the following: Given a propositional formula in conjunctive normal form, can one find an assignment to variables such that there is exactly only 1 literal that is true in every clause, while the other literals are false. The decision problem XSAT is known to be NP-complete. Crescenzi and Rossi [Pierluigi Crescenzi and Gianluca Rossi, 2002] introduced the variant where one searches for a pair of two solutions of an X3SAT instance with maximal Hamming Distance among them, that is, one wants to identify the largest number k such that there are two solutions of the instance with Hamming Distance k. Dahllöf [Vilhelm Dahllöf, 2005; Vilhelm Dahllöf, 2006] provided an algorithm using branch and bound method for Max Hamming Distance XSAT in O(1.8348^n); Fu, Zhou and Yin [Linlu Fu and Minghao Yin, 2012] worked on a more specific problem, the Max Hamming Distance X3SAT, and found for this problem an algorithm with runtime O(1.6760^n). In this paper, we propose an exact exponential algorithm to solve the Max Hamming Distance XSAT problem in O(1.4983^n) time. Like all of them, we will use the branch and bound technique alongside a newly defined measure to improve the analysis of the algorithm.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Design and analysis of algorithms
##### Keywords
• XSAT
• Measure and Conquer
• DPLL
• Exponential Time Algorithms

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