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

By Susanne Albers, Arindam Khan, Leon Ladewig. 2020

40 citations recorded by the source corpus.

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).

Subfields: Computer Networks and Communications, Industrial and Manufacturing Engineering, Optimization & Training

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computer Networks and Communications

1. Simon Lacoste-Julien
2. Pascal Vincent
3. Hugo Berard
4. William L. Hamilton
5. Gauthier Gidel

In Industrial and Manufacturing Engineering

1. Simon Lacoste-Julien
2. Pascal Vincent
3. François Soumis

In Optimization & Training

1. Gauthier Gidel
2. Pascal Vincent
3. Hugo Berard
4. Simon Lacoste-Julien
5. Andjela Mladenovic
6. Avishek Joey Bose
7. William L. Hamilton
8. Peter Richtárik

Citations in the corpus

• Online Adversarial Attacks (2021, methodology): “[2007] and the recent Single-Ref algorithm Albers et al. [2019]. In a nutshell, these online algorithm consists of two phases—a sampling phase followed by a selection phase—and an optimal stopping point t (threshold) that is used by the algorithm to transition between the phases.”
• New Results for the $k$-Secretary Problem (2020, unknown): “the knapsack secretary problem [7–9].”