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Integer programming with 2-variable equations and 1-variable inequalities

Manuel Bodirsky, Gustav Nordh, Timo von Oertzen
2009 Information Processing Letters  
We present an efficient algorithm to find an optimal integer solution of a given system of 2-variable equalities and 1-variable inequalities with respect to a given linear objective function.  ...  Our algorithm has worst-case running time in O(N 2 ) where N is the number of bits in the input.  ...  Again it should be clear (by similar reasoning as in the previous sections) that the necessary computations can be done in O(N 2 ) time.  ... 
doi:10.1016/j.ipl.2009.01.025 fatcat:vdbn6o2blfgjvafq6dbr2q2uwa

A Novel Alternative Algorithm for Solving Linear Integer Programming Problems with Four Variables

2021 European Journal of Science and Technology  
In this paper, new iterative method is proposed based on parametrization for solving Integer Linear Programming (ILP) problems with four variables and an algorithm is provided.  ...  Our method, which is better than the cutting plane method and branch and bound methods in solving ILP problems with four variables, can be easily applied regardless of the number of constraints.  ...  Inequalities with respect to the point (𝑦 1 , 𝑦 3 ) 𝑥 1 Avrupa Bilim ve Is there at least one integer point (𝑦 1 , 𝑦 3 ) satisfyin g the inequality? Is 𝑦 2 an integer?  ... 
doi:10.31590/ejosat.1020212 fatcat:sjaqmcpjmbe2hjco6umxuza7um


Shinto K.G .
2013 International Journal of Research in Engineering and Technology  
The two main proposals for solving integer linear programming problems are those of Gomory [1] and Land and Doig [2 . The method of Gomory starts with the simplex solution.  ...  This elimination process can now be repeated with 2, 3' etc., until finally, one arrives at a set of Nn-2 inequalities involving a -single variable n-1.  ... 
doi:10.15623/ijret.2013.0207012 fatcat:qfzo5hjonjgzdktcmsp3laigua

Parametric Strategy Iteration [article]

Thomas M. Gawlitza, Martin D. Schwarz, Helmut Seidl
2014 arXiv   pre-print
Parametric strategy iteration for systems of integer equations allows to construct parametric integer interval analysis as well as parametric analysis of differences of integer variables.  ...  It thus provides a general technique to realize precise parametric program analysis if numerical properties of integer variables are of concern.  ...  We thank Stefan Barth (LMU) for Example 4, and Jan Reineke (Universität des Saarlandes) for useful discussions.  ... 
arXiv:1406.5457v1 fatcat:rake6of3ovg5fgbzgkt6nn2mny

Propagating dense systems of integer linear equations

Thibaut Feydy, Peter J. Stuckey
2007 Proceedings of the 2007 ACM symposium on Applied computing - SAC '07  
We show on standard integer benchmarks how these new propagators can substantially improve propagation performance, in terms of strength of propagation and speed.  ...  In interval propagation approaches to solving non-linear constraints over reals it is common to build stronger propagators from systems of linear equations.  ...  and n variables will result in a system of m equations, each with (n − m) + 1 non quasi-zero coefficients.  ... 
doi:10.1145/1244002.1244075 dblp:conf/sac/FeydyS07 fatcat:ckstu2cpezcnpd5nl7fvjo2dn4

DIV, FLOOR, CEIL, MOD and STEP Functions in Nested Loop Programs and Linearly Bounded Lattices [chapter]

P.C. Held, A.C.J. Kienhuis
1995 Algorithms and Parallel VLSI Architectures III  
The nested loop programs may contain the integer operators: integer division, floor, ceil, and modulo, in expressions and the stride, or step size, of for loops may be greater than one.  ...  We will show the relation between the integer division operators in the SAP and linearly bounded lattices in the corresponding DG.  ...  The domain of lattice offsets is formed by a polytope in variables of à . The polytope is characterized by matrix ½ and inequalities of the variables standing for the integer divisions.  ... 
doi:10.1016/b978-044482106-5/50024-7 fatcat:p7dbrrolsbeuvdmhf3my22t7zm

Geometric Programming with Discrete Variables Subject to Max-Product Fuzzy Relation Constraints

Zejian Qin, Bingyuan Cao, Shu-Cherng Fang, Xiao-Peng Yang
2018 Discrete Dynamics in Nature and Society  
The problem of geometric programming subject to max-product fuzzy relation constraints with discrete variables is studied.  ...  We proposed a 0-1 mixed integer linear programming model and adopted the branch-and-bound scheme to solve the problem. Numerical experiments confirm that the proposed solution method is effective.  ...  Acknowledgments This work has been supported by the Nature Science Foundation of Guangdong Province (nos. 2016A030307037, 2016A030313552, and QD20171001).  ... 
doi:10.1155/2018/1610349 fatcat:cbv2o5ymfzeebpau6gfntrtcne

A Sequent Calculus for Integer Arithmetic with Counterexample Generation

Philipp Rümmer
2007 Conference on Automated Deduction  
There are four main components: a complete procedure for linear equations, a complete procedure for linear inequalities, an incomplete procedure for nonlinear (polynomial) equations, and an incomplete  ...  The method is tailored to Java program verification and meant to be used both as a supporting procedure and simplifier during interactive verification and as an automated tool for discharging (ground)  ...  I want to thank Wolfgang Ahrendt and Richard Bubel for many inspiring discussions and comments on this paper. Thanks are also due to the anonymous referees for helpful comments.  ... 
dblp:conf/cade/Rummer07 fatcat:t23aqt7sfbfobdoul6fha32iqm

The Elimination of Integer Variables

H. P. Williams
1992 Journal of the Operational Research Society  
one all-integer inequality row (2) with entries in other rows (2) all being 0, negative or +1; or (iii) it has an entry —1 in at least one all-integer inequality row (2) with entries in other rows (2)  ...  Also, we can divide through these inequalities by the relevant non-zero coefficients giving 1 n iq j=0 JS#q 1 - 1 (> - > au 2X, kel Ang j=0 J#q together with (1) and inequalities (2) in which x, has no  ... 
doi:10.1057/jors.1992.65 fatcat:dkogfv5tivh5vftn66vazk23qq

Integer Programming and Pricing

Ralph E. Gomory, William J. Baumol
1960 Econometrica  
method, starting with the integer inequalities in n a*,o + E a*, (-x>) (aO,j integers), J=1 one introduces slack variables x', one for each inequality, converting them into equations n (i.2) x/ a= a*  ...  It is derived from an equation n ai,o + I ai,1 (-t1)1=1 where the variables, t, are either x's or slack variables of the original problem.  ...  Now the expression for the variables in terms of the non-basic set x' is unique, so all the coefficients must be identical with those obtained in solving the n inequalities of an ordinary linear programming  ... 
doi:10.2307/1910130 fatcat:e2fkqkjvj5g6vmgziipekr7raa

A Novel Alternative Algorithm for Solving Integer Linear Programming Problems Having Three Variables

Kadriye Simsek Alan
2020 Cybernetics and Information Technologies  
This method, which is better than the cutting plane and branch boundary method, can be applied to pure integer linear programming problems with m linear inequality constraints, a linear objective function  ...  with three variables.  ...  It has an integer solution if and only if ; also, if a linear Diophantine equation has an integer solution; then there will be infinitely many solution for this equation [1] .  ... 
doi:10.2478/cait-2020-0045 fatcat:urcnikbz5fewdih6c5mzgaaxli

Propagating systems of dense linear integer constraints

Thibaut Feydy, Peter J. Stuckey
2008 Constraints  
In interval propagation approaches to solving nonlinear constraints over reals it is common to build stronger propagators from systems of linear equations.  ...  In a similar fashion we present an interval Fourier elimination preconditioning technique to generate redundant linear constraints from a system of linear inequalities.  ...  Example 5 Consider the integer variables x1, x2, x3, with the domains d1 = d2 = d3 = [−10, 10], and the following linear constraints: C1 : 2 4 5 3 4 −1 22 11 2 3 5 2 4 x1 x2 x3 3 5 = 2 4 0 7 −2 3  ... 
doi:10.1007/s10601-008-9049-9 fatcat:x3sx3tpw4bdxnatnoychq7r7gy

The general form of 0–1 programming problem based on DNA computing

Yin ZhiXiang, Zhang Fengyue, Xu Jin
2003 Biosystems (Amsterdam. Print)  
In this paper, we solved the general form of 0-1 programming problem with fluorescence labeling techniques based on surface chemistry by attempting to apply DNA computing to a programming problem.  ...  Up to now, many accomplishments have been made to improve its performance and increase its reliability.  ...  They also wish to thank an anonymous referee of this paper who provided many useful and constructive suggestions for the improvement of this paper.  ... 
doi:10.1016/s0303-2647(03)00053-4 pmid:12753938 fatcat:qztoa5vmkzahrif33hzpjqrcuu

Mixed integer polynomial programming

Vivek Dua
2015 Computers and Chemical Engineering  
The mixed integer polynomial programming problem is reformulated as a multi-parametric programming problem by relaxing integer variables as continuous variables and then treating them as parameters.  ...  solution as a function of the relaxed integer variables.  ...  Acknowledgments This paper is dedicated to Professor Ignacio Grossmann in celebration of his 65th birthday and in acknowledgement of his contributions to the field of Process Systems Engineering.  ... 
doi:10.1016/j.compchemeng.2014.07.020 fatcat:pt7rpsmdjzfs7px5ba75islje4

Fourier-Motzkin elimination extension to integer programming problems

H.P Williams
1976 Journal of combinatorial theory. Series A  
deal with Integer Programming (IP) problems.  ...  This paper describes how the Fourier-Motzkin Elimination Method, which can be used for solving Linear Programming Problems, can be extended to deal with Integer Programming Problems.  ...  with Integer Programming Problems.  ... 
doi:10.1016/0097-3165(76)90055-8 fatcat:aeynivmfwnd3rjo3k2ty6rcck4
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