On Strong NP-Completeness of Rational Problems

The computational complexity of the partition, 0-1 subset sum, unbounded subset sum, 0-1 knapsack and unbounded knapsack problems and their multiple variants were studied in numerous papers in the past where all the weights and profits were assumed to be integers. We re-examine here the computational complexity of all these problems in the setting where the weights and profits are allowed to be any rational numbers. We show that all of these problems in this setting become strongly NP-complete and, as a result, no pseudo-polynomial algorithm can exist for solving them unless P=NP. Despite this result we show that they all still admit a fully polynomial-time approximation scheme.

[1]  Christelle Guéret,et al.  A new lower bound for the open‐shop problem , 1999, Ann. Oper. Res..

[2]  Sven Schewe,et al.  Optimal Control for Simple Linear Hybrid Systems , 2016, 2016 23rd International Symposium on Temporal Representation and Reasoning (TIME).

[3]  D. S. Johnson,et al.  On Knapsacks, Partitions, and a New Dynamic Programming Technique for Trees , 1983, Math. Oper. Res..

[4]  David S. Johnson,et al.  `` Strong '' NP-Completeness Results: Motivation, Examples, and Implications , 1978, JACM.

[5]  David Pisinger.  An exact algorithm for large multiple knapsack problems , 1999, Eur. J. Oper. Res..

[6]  Deeparnab Chakrabarty,et al.  Knapsack Problems , 2008 .

[7]  Sanjeev Khanna,et al.  A Polynomial Time Approximation Scheme for the Multiple Knapsack Problem , 2005, SIAM J. Comput..

[8]  Sven Schewe,et al.  Optimal Control for Multi-Mode Systems with Discrete Costs , 2017, FORMATS.

[9]  G. B. Mathews.  On the Partition of Numbers , 1896 .

[10]  Thomas J. Schaefer,et al.  The complexity of satisfiability problems , 1978, STOC.

[11]  Han Hoogeveen,et al.  New Lower and Upper Bounds for Scheduling Around a Small Common Due Date , 1994, Oper. Res..

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

[13]  Paolo Toth,et al.  Knapsack Problems: Algorithms and Computer Implementations , 1990 .

[14]  J. Rosser,et al.  Approximate formulas for some functions of prime numbers , 1962 .

[15]  Oscar H. Ibarra,et al.  Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems , 1975, JACM.

[16]  Craig A. Tovey,et al.  A simplified NP-complete satisfiability problem , 1984, Discret. Appl. Math..

[17]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[18]  Ulrich Pferschy,et al.  The Knapsack Problem with Conflict Graphs , 2009, J. Graph Algorithms Appl..

[19]  Vijay V. Vazirani,et al.  Approximation Algorithms , 2001, Springer Berlin Heidelberg.

[20]  Michael Wooldridge,et al.  On the computational complexity of weighted voting games , 2009, Annals of Mathematics and Artificial Intelligence.