Exact and greedy solutions of the knapsack problem: the ratio of values of objective functions

  • Authors:

  • A. A. Korbut;I. Kh. Sigal

  • Affiliations:

  • Institute for Economics and Mathematics, Russian Academy of Sciences, St. Petersburg, Russia 191187 and Computing Center, Russian Academy of Sciences, Moscow, Russia 119991;Institute for Economics and Mathematics, Russian Academy of Sciences, St. Petersburg, Russia 191187 and Computing Center, Russian Academy of Sciences, Moscow, Russia 119991

  • Venue:
  • Journal of Computer and Systems Sciences International
  • Year:
  • 2010

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Abstract

Ratios 驴 of the values of objective functions of optimal Boolean (or integer) to the values of greedy solutions for the knapsack problem are considered. The relationship of the parameter 驴 with the ratio Δ of the values of objective functions for the optimal solution of linear relaxation to the values of optimal integer solution was found. Two-sided estimates for 驴 and Δ were obtained. A computational experiment was conducted to investigate the ratio of 驴 of problems of one- and two-dimensional knapsack problems with Boolean variables. A hypothesis on asymptotic behavior of the ratio 驴 with growth of the number of problem variables was formulated.