Knapsack problems: algorithms and computer implementations
Knapsack problems: algorithms and computer implementations
Lifting cover inequalities for the precedence-constrained knapsack problem
Discrete Applied Mathematics
Simple but efficient approaches for the collapsing knapsack problem
Discrete Applied Mathematics
Algorithmic geometry
A fast algorithm for strongly correlated knapsack problems
Discrete Applied Mathematics
Algorithms for the bounded set-up knapsack problem
Discrete Optimization
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The Integer Knapsack Problem with Set-up Weights (IKPSW) is a generalization of the classical Integer Knapsack Problem (IKP), where each item type has a set-up weight that is added to the knapsack if any copies of the item type are in the knapsack solution. The k-item IKPSW (kIKPSW) is also considered, where a cardinality constraint imposes a value k on the total number of items in the knapsack solution. IKPSW and kIKPSW have applications in the area of aviation security. This paper provides dynamic programming algorithms for each problem that produce optimal solutions in pseudo-polynomial time. Moreover, four heuristics are presented that provide approximate solutions to IKPSW and kIKPSW. For each problem, a Greedy heuristic is presented that produces solutions within a factor of 1/2 of the optimal solution value, and a fully polynomial time approximation scheme (FPTAS) is presented that produces solutions within a factor of 驴 of the optimal solution value. The FPTAS for IKPSW has time and space requirements of O(nlog驴n+n/驴 2+1/驴 3) and O(1/驴 2), respectively, and the FPTAS for kIKPSW has time and space requirements of O(kn 2/驴 3) and O(k/驴 2), respectively.