Improved Online Algorithms for Knapsack and GAP in the Random Order Model

Abstract

The knapsack problem is one of the classical problems in combinatorial optimization: Given a set of items, each specified by its size and profit, the goal is to find a maximum profit packing into a knapsack of bounded capacity. In the online setting, items are revealed one by one and the decision, if the current item is packed or discarded forever, must be done immediately and irrevocably upon arrival. We study the online variant in the random order model where the input sequence is a uniform random permutation of the item set. We develop a randomized (1/6.65)-competitive algorithm for this problem, outperforming the current best algorithm of competitive ratio 1/8.06 (Kesselheim et al. in SIAM J Comput 47(5):1939–1964, 2018). Our algorithm is based on two new insights: We introduce a novel algorithmic approach that employs two given algorithms, optimized for restricted item classes, sequentially on the input sequence. In addition, we study and exploit the relationship of the knapsack problem to the 2-secretary problem. The generalized assignment problem (GAP) includes, besides the knapsack problem, several important problems related to scheduling and matching. We show that in the same online setting, applying the proposed sequential approach yields a (1/6.99)-competitive randomized algorithm for GAP. Again, our proposed algorithm outperforms the current best result of competitive ratio 1/8.06 (Kesselheim et al. in SIAM J Comput 47(5):1939–1964, 2018).

Improved Online Algorithm for Fractional Knapsack in the Random Order Model

“…α) and either executing the given algorithm or the famous secretary algorithm, where the first n/e items are discarded and then the first item that is better than the best item seen so far is chosen. Together with the result of [1], this led to the previously best known competitive ratio of 9.37 for the online fractional knapsack problem. In this paper, we cut the competitive ratio by more than half, namely to 4.39.…”

“…Kesselheim et al [24] developed an 8.06-competitive randomized algorithm for the generalized assignment problem, which generalizes to a setting with multiple knapsacks of different capacities. Albers et al [1] achieved the currently best known upper bound of 6.65. The online fractional variant under adversarial arrivals was first considered in [30].…”

“…The o(1)-term vanishes asymptotically with respect to the number of items in the input instance. In a series of papers [1,5,24], the competitive ratio for online knapsack has been improved to 6.65, achieved by a randomized algorithm in [1]. The online version of the fractional knapsack problem has not received much attention in literature as of yet.…”

“…In this paper, we cut the competitive ratio by more than half, namely to 4.39. To this end, we observed that the algorithm in [1] already contains a secretary algorithm on its own and that small and large items are treated separately due to the nature of the integral knapsack solution. Instead of treating it as a black box, relaxing the integrality constraints allows us to unify the small-item and large-item cases and output fractional values instead of binary ones, thereby achieving a better competitive ratio.…”

Machine Covering in the Random-Order Model

“…In this model, jobs are still chosen worst possible by the adversary but they are presented to the online algorithm in a uniformly random order. The random-order model derives from the Secretary Problem [13,35] and has been applied to a wide variety of problems such as generalized Secretary problems [32,8,33,18,9], Scheduling problems [39,38,2,3], Packing problems [30,31,5,4], Facility Location problems [36] and Convex Optimization problems [25] among others. See also [26] for a survey chapter.…”