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Multiple knapsack problem full version#
Multiple knapsack problem free#

Multiple knapsack problem full version#

Finally, in the full version of this paper, we illustrate the proposed algorithm via trace-based experiments of EV charging. Moreover, our analysis provides a novel approach to online algorithm design based on an instance-dependent primal-dual analysis that connects the identification of worst-case instances to the design of algorithms. We introduce a new algorithm that achieves a competitive ratio within an additive factor of the best achievable competitive ratios for the general problem and matches or improves upon the best-known competitive ratio for special cases in the knapsack and one-way trading literatures. This problem generalizes variations of the knapsack problem and of the one-way trading problem that have previously been treated separately, and additionally finds application to the real-time control of electric vehicle (EV) charging. Let’s define a state dp where i denotes that we are considering the i th element, j denotes the current weight filled, and k denotes the number of items filled until now.įor every state dp, the profit is either that of the previous state (when the current state is not included) or the profit of the current item added to that of the previous state (when the current item is selected).We introduce and study a general version of the fractional online knapsack problem with multiple knapsacks, heterogeneous constraints on which items can be assigned to which knapsack, and rate-limiting constraints on the assignment of items to knapsacks. Let us consider a 3-dimensional table dp, where N is the number of elements, W is the maximum weight capacity and K is the maximum number of items allowed in the knapsack. Either way, the optimal substructure property is satisfied. For a particular subset of z elements, the solution for (z+1) th element can either have a solution corresponding to the z elements or the (z+1) th element can be added if it doesn’t exceed the knapsack constraints.

Optimal substructure: Overall, each item has only two choices, either it can be included in the solution or denied.

Hence there will be overlapping solutions. In the second iteration we have (1, 2) and so on where (1) and (2) are recalculated.

Overlapping sub-problems: When the recursive solution is tried, 1 item is added first and the solution set is (1), (2), …(n).

Let us first verify that the conditions of DP are still satisfied. Recommended: Please try your approach on first, before moving on to the solution.Īpproach: The dynamic programming approach is preferred over the general recursion approach.

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Multiple knapsack problem free#

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