Improved exponential time lower bound of Knapsack problem under BT model

  • Authors:

  • Xin Li;Tian Liu;Han Peng;Liyan Qian;Hongtao Sun;Jin Xu;Ke Xu;Jiaqi Zhu

  • Affiliations:

  • School of EECS, Beijing University, Beijing, China;School of EECS, Beijing University, Beijing, China;School of EECS, Beijing University, Beijing, China;School of EECS, Beijing University, Beijing, China;School of EECS, Beijing University, Beijing, China;School of EECS, Beijing University, Beijing, China;Department of Computer Science, Beihang University, Beijing, China;School of EECS, Beijing University, Beijing, China

  • Venue:
  • TAMC'07 Proceedings of the 4th international conference on Theory and applications of models of computation
  • Year:
  • 2007

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Abstract

M. Alekhnovich et al. have recently proposed a model of algorithms, called BT model, which covers Greedy, Backtracking and Simple Dynamic Programming algorithms and can be further divided into three kinds of fixed, adaptive and fully adaptive ones, and have proved exponential time lower bounds of exact and approximation algorithms under adaptive BT model for Knapsack problem which are about Ω(20.5n/√n) and Ω((1/Ɛ)0.315)(for approximation ratio 1 - Ɛ), respectively (M. Alekhovich, A. Borodin, J. Buresh-Oppenheim, R. Impagliazzo, A. Magen, and T. Pitassi, Toward a Model for Backtracking and Dynamic Programming, Proceedings of Twentieth Annual IEEE Conference on Computational Complexity, pp308-322, 2005). In this short note, we slightly improve their lower bounds to Ω(20.66n/√n) and Ω((1/Ɛ)0.420), respectively, through more complicated combinatorial arguments, and propose as an open question what is the best achievable lower bound for Knapsack problem under the adaptive BT model.