An algebraic expression of the number partitioning problem

Abstract In this paper we investigate the number partitioning problem, using the tropical semiring (max-plus algebra). We show that the problem is reduced to deciding whether one of integers is a solution of a tropical analogue of algebraic equations with coefficients composed of other integers. For n up to 6 we derive concretely and explicitly the equation and its solution set. The derivation requires only routine algebraic computations, so can be applied for n larger than 6. Our approach based on max-plus algebra reveals the mathematical structure of the problem and provides a new view point for the P versus NP problem, since the problem is well-known to be NP-complete.

[1]  Howard Straubing,et al.  A combinatorial proof of the Cayley-Hamilton theorem , 1983, Discret. Math..

[2]  Adi Shamir,et al.  A T=O(2n/2), S=O(2n/4) Algorithm for Certain NP-Complete Problems , 1981, SIAM J. Comput..

[3]  Jean-Louis Bouquard,et al.  Two-machine flow shop scheduling problems with minimal and maximal delays , 2006, 4OR.

[4]  Rainer E. Burkard,et al.  Finding all essential terms of a characteristic maxpolynomial , 2003, Discret. Appl. Math..

[5]  Rainer E. Burkard,et al.  Max algebra and the linear assignment problem , 2003, Math. Program..

[6]  Cecilia R. Aragon,et al.  Optimization by Simulated Annealing: An Experimental Evaluation; Part II, Graph Coloring and Number Partitioning , 1991, Oper. Res..

[7]  Eugenii Shustin,et al.  Tropical Algebraic Geometry , 2007 .

[8]  R. A. Cuninghame-Green,et al.  Maxpolynomial equations , 1995, Fuzzy Sets Syst..

[9]  S. Mertens.  Phase Transition in the Number Partitioning Problem , 1998, cond-mat/9807077.

[10]  Richard E. Korf,et al.  A Complete Anytime Algorithm for Number Partitioning , 1998, Artif. Intell..

[11]  Geert Jan Olsder,et al.  Synchronization and Linearity: An Algebra for Discrete Event Systems , 1994 .

[12]  Fred W. Glover,et al.  A new modeling and solution approach for the number partitioning problem , 2005, Adv. Decis. Sci..

[13]  Tae-Eog Lee.  Stable Earliest Starting Schedules for Periodic Job Shops: a Linear System Approach , 1994 .

[14]  Geert Jan Olsder,et al.  Cramer and Cayley-Hamilton in the max algebra , 1988 .

[15]  Ludwig Elsner,et al.  On the power method in max algebra , 1999 .

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

[17]  Raymond Cuninghame-Green,et al.  The characteristic maxpolynomial of a matrix , 1983 .

[18]  P. Butkovic.  Max-linear Systems: Theory and Algorithms , 2010 .

[20]  Richard M. Karp,et al.  The Differencing Method of Set Partitioning , 1983 .

[21]  Cecilia R. Aragon,et al.  Optimization by Simulated Annealing: An Experimental Evaluation; Part I, Graph Partitioning , 1989, Oper. Res..

[22]  Li-Hui Tsai,et al.  Asymptotic Analysis of an Algorithm for Balanced Parallel Processor Scheduling , 1992, SIAM J. Comput..

[23]  Richard M. Karp,et al.  Reducibility Among Combinatorial Problems , 1972, 50 Years of Integer Programming.

[24]  P. Binding,et al.  A generalized eigenvalue problem in the max algebra , 2007 .

[25]  Katsuhiro Nishinari,et al.  Applications of max-plus algebra to flow shop scheduling problems , 2018, Discret. Appl. Math..

[26]  Claire Hanen,et al.  A Study of the Cyclic Scheduling Problem on Parallel Processors , 1995, Discret. Appl. Math..

[27]  Brian Hayes,et al.  The Easiest Hard Problem , 2002, American Scientist.

[28]  Ellis Horowitz,et al.  Computing Partitions with Applications to the Knapsack Problem , 1974, JACM.

[29]  J. Marks,et al.  Easily searched encodings for number partitioning , 1996 .

[30]  Bernard Giffler.  Schedule algebra: A progress report , 1968 .

[31]  Gunnar Carlsson,et al.  Symmetric and r-symmetric tropical polynomials and rational functions , 2014, 1405.2268.

[32]  Alexander K. Hartmann,et al.  Solution-space structure of (some) optimization problems , 2007, 0711.3912.

[33]  Raymond A. Cuninghame-Green,et al.  Projections in minimax algebra , 1976, Math. Program..