On the rectangular knapsack problem: approximation of a specific quadratic knapsack problem

In this article, we introduce the rectangular knapsack problem as a special case of the quadratic knapsack problem consisting in the maximization of the product of two separate knapsack profits subject to a cardinality constraint. We propose a polynomial time algorithm for this problem that provides a constant approximation ratio of 4.5. Our experimental results on a large number of artificially generated problem instances show that the average ratio is far from theoretical guarantee. In addition, we suggest refined versions of this approximation algorithm with the same time complexity and approximation ratio that lead to even better experimental results.

[1]  Alain Billionnet,et al.  An exact method based on Lagrangian decomposition for the 0-1 quadratic knapsack problem , 2004, Eur. J. Oper. Res..

[2]  P. Hammer,et al.  Quadratic knapsack problems , 1980 .

[3]  Alain Billionnet,et al.  Linear programming for the 0–1 quadratic knapsack problem , 1996 .

[4]  Hans Kellerer,et al.  Fully Polynomial Approximation Schemes for a Symmetric Quadratic Knapsack Problem and its Scheduling Applications , 2010, Algorithmica.

[5]  Xin-She Yang,et al.  Introduction to Algorithms , 2021, Nature-Inspired Optimization Algorithms.

[6]  Zhou Xu,et al.  A strongly polynomial FPTAS for the symmetric quadratic knapsack problem , 2012, Eur. J. Oper. Res..

[7]  J. Rhys.  A Selection Problem of Shared Fixed Costs and Network Flows , 1970 .

[8]  David Pisinger,et al.  The quadratic knapsack problem - a survey , 2007, Discret. Appl. Math..

[9]  David Pisinger,et al.  Solution of Large Quadratic Knapsack Problems Through Aggressive Reduction , 2007, INFORMS J. Comput..

[10]  Alain Billionnet,et al.  A new upper bound for the 0-1 quadratic knapsack problem , 1999, Eur. J. Oper. Res..

[11]  Panos M. Pardalos,et al.  Quadratic Assignment Problem , 1997, Encyclopedia of Optimization.

[12]  Gerhard J. Woeginger,et al.  The quadratic 0-1 knapsack problem with series-parallel support , 2002, Oper. Res. Lett..

[13]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[14]  Lothar Thiele,et al.  Multiobjective Optimization Using Evolutionary Algorithms - A Comparative Case Study , 1998, PPSN.

[15]  R. Weismantel,et al.  A Semidefinite Programming Approach to the Quadratic Knapsack Problem , 2000, J. Comb. Optim..

[16]  Carlos M. Fonseca,et al.  Hypervolume Subset Selection in Two Dimensions: Formulations and Algorithms , 2016, Evolutionary Computation.

[17]  George L. Nemhauser,et al.  Min-cut clustering , 1993, Math. Program..

[18]  Arie M. C. A. Koster,et al.  Combinatorial Optimization on Graphs of Bounded Treewidth , 2008, Comput. J..

[19]  Philippe Michelon,et al.  0-1 Quadratic Knapsack Problems: An Exact Approach Based on a t-Linearization , 2012, SIAM J. Optim..

[20]  Richard Taylor,et al.  Approximation of the Quadratic Knapsack Problem , 2015, Oper. Res. Lett..

[21]  Paolo Toth,et al.  Exact Solution of the Quadratic Knapsack Problem , 1999, INFORMS J. Comput..

[22]  Christoph Witzgall.  Mathematical methods of site selection for Electronic Message Systems (EMS) , 1975 .