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Online knapsack problem is considered, where items arrive in a sequentialfashion that have two attributes; value and weight. Each arriving item has tobe accepted or rejected on its arrival irrevocably. The objective is tomaximize the sum of the value of the accepted items such that the sum of theirweights is below a budget/capacity. Conventionally a hard budget/capacityconstraint is considered, for which variety of results are available. In modernapplications, e.g., in wireless networks, data centres, cloud computing, etc.,enforcing the capacity constraint in expectation is sufficient. With thismotivation, we consider the knapsack problem with an expected capacityconstraint. For the special case of knapsack problem, called the secretaryproblem, where the weight of each item is unity, we propose an algorithm whoseprobability of selecting any one of the optimal items is equal to $1-1/e$ andprovide a matching lower bound. For the general knapsack problem, we propose analgorithm whose competitive ratio is shown to be $1/4e$ that is significantlybetter than the best known competitive ratio of $1/10e$ for the knapsackproblem with the hard capacity constraint.

Online Knapsack Problem under Expected Capacity Constraint