Sequential knapsack problem
The authors show how this special class of knapsack problems can be solved in O(n) time after sorting the profit-to-weight ratios p{sub j}/w{sub j}. Constraints with this integer-multiples property arise in the lexicographic ordering of 0-1 vectors, and with bounded-integer variables in capacity expansion problems for high speed networks. A sequential knapsack relaxation of the 0-1 knapsack problem can be derived for every sequence of integers 1 = d{sub 1}, d{sub 2}, ..., d{sub r} such that d{sub 1}{vert_bar}d{sub 2}{vert_bar}{center_dot}{center_dot}{center_dot}{vert_bar}d{sub r}. When d{sub k} = 2{sup k-1} for k = 1, 2, ..., r the relaxation is a 0-1 sequential knapsack problems, but in general the bounds on the variables are determined by expanding the weights according to the sequences d{sub 1}{vert_bar}d{sub 2}{vert_bar}{center_dot}{center_dot}{center_dot}{vert_bar}d{sub r}. The resulting bounded-integer sequential knapsack problem can be solved in O({Sigma}{sub j=1}{sup n} log{sub 2}w{sub j}) time, yielding a bound for the 0-1 knapsack problem. The authors show that this relaxation can also be used to generate valid inequalities (cutting planes) for general 0-1 integer programming problems. The method uses the 0-1 knapsack problems obtained by dropping all but one of the constraints of the 0-1 integer programming problem, but unlike the case of minimal cover inequalities the separation problem for the resulting class of inequalities can be solved in polynomial time (it reduces to the separation problem for a bounded-integer sequential knapsack problem).
- OSTI ID:
- 36114
- Report Number(s):
- CONF-9408161--
- Country of Publication:
- United States
- Language:
- English
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