Abstract
Two sets of numbers are generated to determine the feasibility and periodicity for an equality-constraint knapsack problem. By aid of the two sets, it is shown that for only a finite number of right hand side numbers the knapsack problem is hard. A novel method is given for solving such hard knapsack problems, of which the complexities of a main part does not depend on the number of variables involved, provided that the constraint coefficients are bounded. A method for representing all the optimal solutions of the knapsack problem for all right hand side numbers is further shown. A similar theory is also developed for an equality-constraint integer linear program with mixed signs in the constraint coefficients.
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