9/19/2023 0 Comments Knapsack approximation algorithmIbarra, O.H., Kim, C.E.: Fast approximation algorithms for the knapsack and sum of subset problems. Cambridge University Press, New York (2011) Williamson, D.P., Shmoys, D.B.: The Design of Approximation Algorithms. Freeman, New York (1979)Ĭhekuri, C., Khanna, S.: A polynomial time approximation scheme for the multiple knapsack problem. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Nip, K., Wang, Z., Xing, W.: A study on several combination problems of classic shop scheduling and shortest path. Nip, K., Wang, Z., Talla Nobibon, F., Leus, R.: A combination of flow shop scheduling and the shortest path problem. Nip, K., Wang, Z., Xing, W.: Combinations of some shop scheduling problems and the shortest path problem: complexity and approximation algorithms. Nip, K., Wang, Z.: Combination of two-machine flow shop scheduling and shortest path problems. Wang, Z., Cui, Z.: Combination of parallel machine scheduling and vertex cover. Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. We finally provide a polynomial time approximation scheme for this problem. Particularly, the first one is a \(\frac - \epsilon \) for any small enough \(\epsilon >0\). We mainly propose three approximation algorithms for it. It is a generalization of the multiple knapsack problem, and hence is strongly NP-hard. This problem is motivated by various practical applications, e.g., in logistics. The total profit of the problem is determined by the items that are selected and packed into the container within some packed box, and the objective is to maximize the total profit. The first phase is to select some items to pack into the boxes, and the second phase is to select the boxes (each includes some packed items) to pack into the container. Each item has a size and profit, each box has a size and the container has a capacity. In this problem, we have a set of items, several small knapsacks called boxes, and a large knapsack called container. We consider a natural generalization of the knapsack problem and the multiple knapsack problem, which has two phases of packing decisions.
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