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).

Highlights

  • Many real-world problems can be considered resource allocation problems

  • The dispatcher seeks for a maximum profit packing fulfilling the capacity constraint

  • We study online variants of the knapsack problem and generalized assignment problem (GAP)

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Summary

Introduction

Many real-world problems can be considered resource allocation problems. For example, consider the loading of a cargo plane with (potential) goods of different weights. The knapsack problem admits no randomized algorithm of bounded competitive ratio in the general online setting [12]. This holds even if only a single item can be packed, as known from the secretary problem [13, 14]. These hardness results are based on a worst-case input presented in adversarial order. The model has been successfully applied to other problem classes including scheduling [19,20,21], packing [22,23,24,25,26], graph problems [27,28,29], facility location [30], budgeted allocation [31], and submodular welfare maximization [32]

Online Knapsack Problem
Online GAP
Online packing LPs
Our Contributions
Roadmap
Knapsack Problem
Bounding Sums by Integrals
Sequential Approach
Large Items
Packing Types
Acceptance Probabilities of Algorithm 2
Analysis
Two‐item Cases
Competitive Ratio
Discussion of other 2‐Secretary Algorithms
Small Items
Algorithm
Extension to GAP
Large Options
Small Options
Remark
Proof of Theorem 2
Full Text
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