Abstract
In this paper, we study the approximation complexity of the Minimum Integral Solution Problem with Preprocessing introduced by Alekhnovich et al. (FOCS, pp. 216–225, 2005). We show that the Minimum Integral Solution Problem with Preprocessing over \(\ell _\infty \) norm (\(\hbox {MISPP}_\infty \)) is NP-hard to approximate to within a factor of \((\log n)^{1/2-\epsilon },\) unless \(\mathbf{NP}\subseteq \mathbf{DTIME}(2^{poly log(n)}).\) This improves on the best previous result. The best result so far gave \(\sqrt{2}-\epsilon \) factor hardness for any \(\epsilon >0.\)
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Acknowledgments
We would like to thank the anonymous referees for their careful readings of the manuscripts and many useful suggestions. Wenbin Chen’s research has been supported by the National Natural Science Foundation of China (NSFC) under Grant No. 11271097, faculties’ starting funding of Guangzhou University, the Research Project of Guangzhou Education Bureau under Grant No. 2012A074, the Project IIPL-2011-001 from Shanghai Key Laboratory of Intelligent Information Processing, and the Project KFKT2012B01 from State Key Laboratory for Novel Software Technology, Nanjing University. Lingxi Peng’s research has been also partly supported by the National Natural Science Foundation of China (NSFC) under Grant No. 61100150 and the Research Project of Guangzhou Education Bureau under Grant No. 2012A077. Jianxiong Wang’s research was partially supported under Guangzhou City Council’s Science and Technology Projects funding scheme (Project No. 12C42011622) and under Guangdong Provincial Education Department’s Yumiao early career researchers development funding scheme (2012WYM0105 and 2012LYM0105) and the Research Project of Guangzhou Education Bureau under Grant No. 2012A143. Fufang Li’s research was partially supported by Natural Science Foundation of Guangdong Province of China under Grant No. S2011040003843. Maobin Tang’s research has been also partly supported under Guangdong Province’s Science and Technology Projects under Grant Nos. 2011B020313023 and 2012A020602065 and the Research Project of Guangzhou Education Bureau under Grant No. 2012A075. Wei Xiong’s research was partially supported by Natural Science Foundation of Guangdong Province of China under Grant No. 10451009101004574.
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Chen, W., Peng, L., Wang, J. et al. An improved lower bound for approximating the Minimum Integral Solution Problem with Preprocessing over \(\ell _\infty \) norm. J Comb Optim 30, 447–455 (2015). https://doi.org/10.1007/s10878-013-9646-4
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DOI: https://doi.org/10.1007/s10878-013-9646-4