Integer Programming

Integer programming is an expressive framework for modeling and solving discrete optimization problems that arise in a variety of contexts in the engineering sciences. Integer programming representations work with implicit algebraic constraints (linear equations and inequalities on integer valued variables) to capture the feasible set of alternatives, and linear objective functions (to minimize or maximize over the feasible set) that specify the criterion for defining optimality. This algebraic approach permits certain natural extensions of the powerful methodologies of linear programming to be brought to bear on combinatorial optimization and on fundamental algorithmic questions in the geometry of numbers.

[1]  Klaus Truemper,et al.  Alpha-balanced graphs and matrices and GF(3)-representability of matroids , 1982, J. Comb. Theory, Ser. B.

[2]  László Lovász,et al.  Factoring polynomials with rational coefficients , 1982 .

[3]  Philip Wolfe,et al.  Validation of subgradient optimization , 1974, Math. Program..

[4]  Mihalis Yannakakis,et al.  Expressing combinatorial optimization problems by linear programs , 1991, STOC '88.

[5]  Manfred W. Padberg,et al.  Perfect zero–one matrices , 1974, Math. Program..

[6]  Ralph E. Gomory,et al.  An algorithm for integer solutions to linear programs , 1958 .

[7]  Richard M. Karp,et al.  On Linear Characterizations of Combinatorial Optimization Problems , 1982, SIAM J. Comput..

[8]  Gérard Cornuéjols,et al.  Ideal 0, 1 Matrices , 1994, J. Comb. Theory, Ser. B.

[9]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[10]  I. Borosh,et al.  Bounds on positive integral solutions of linear Diophantine equations , 1976 .

[11]  H. P. Williams,et al.  Logic-Based Decision Support: Mixed Integer Model Formulation , 1989 .

[12]  K. Hoffman,et al.  Large-scale 0–1 linear programming on distributed workstations , 1990 .

[13]  Jeffrey C. Lagarias,et al.  Knapsack Public Key Cryptosystems and Diophantine Approximation , 1983, CRYPTO.

[14]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, Comb..

[15]  Ravi Kannan,et al.  Polynomial Algorithms for Computing the Smith and Hermite Normal Forms of an Integer Matrix , 1979, SIAM J. Comput..

[16]  László Lovász,et al.  Normal hypergraphs and the perfect graph conjecture , 1972, Discret. Math..

[17]  Robert G. Jeroslow,et al.  There Cannot be any Algorithm for Integer Programming with Quadratic Constraints , 1973, Oper. Res..

[18]  Ravi Kannan,et al.  A Polynomial Algorithm for the Two-Variable Integer Programming Problem , 1980, JACM.

[19]  Farid Alizadeh,et al.  Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization , 1995, SIAM J. Optim..

[20]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[21]  V. Chvátal.  On certain polytopes associated with graphs , 1975 .

[22]  P. Camion.  Characterization of totally unimodular matrices , 1965 .

[23]  H. P. Williams.  Experiments in the formulation of integer programming problems , 1974 .

[24]  Sidnie Dresher Feit.  A Fast Algorithm for the Two-Variable Integer Programming Problem , 1984, JACM.

[25]  Martin E. Dyer,et al.  On the Complexity of Computing the Volume of a Polyhedron , 1988, SIAM J. Comput..

[26]  Martin E. Dyer,et al.  A Random Polynomial Time Algorithm for Approximating the Volume of Convex Bodies , 1989, STOC.

[27]  M. Padberg.  Covering, Packing and Knapsack Problems , 1979 .

[28]  Hendrik W. Lenstra,et al.  Integer Programming with a Fixed Number of Variables , 1983, Math. Oper. Res..

[29]  Leslie E. Trotter,et al.  Hermite Normal Form Computation Using Modulo Determinant Arithmetic , 1987, Math. Oper. Res..

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

[31]  Alexander I. Barvinok,et al.  A Polynomial Time Algorithm for Counting Integral Points in Polyhedra when the Dimension Is Fixed , 1993, FOCS.

[32]  David P. Williamson,et al.  .879-approximation algorithms for MAX CUT and MAX 2SAT , 1994, STOC '94.

[33]  A. Shamir.  A polynomial time algorithm for breaking the basic Merkle-Hellman cryptosystem , 1982, FOCS 1982.

[34]  H. D. Ratliff,et al.  Minimum cuts and related problems , 1975, Networks.

[35]  John N. Hooker,et al.  Extended Horn sets in propositional logic , 1991, JACM.

[36]  M. R. Rao,et al.  Odd Minimum Cut-Sets and b-Matchings , 1982, Math. Oper. Res..

[37]  G. Nemhauser,et al.  Integer Programming , 2020 .

[38]  M. Padberg.  Equivalent knapsack‐type formulations of bounded integer linear programs: An alternative approach , 1972 .

[39]  Prabhakar Raghavan,et al.  Randomized rounding: A technique for provably good algorithms and algorithmic proofs , 1985, Comb..

[40]  Clyde L. Monma,et al.  On the Computational Complexity of Integer Programming Problems , 1978 .

[41]  András Frank,et al.  A Weighted Matroid Intersection Algorithm , 1981, J. Algorithms.

[42]  Michael Jünger,et al.  Experiments in quadratic 0–1 programming , 1989, Math. Program..

[43]  J. Edmonds,et al.  A Min-Max Relation for Submodular Functions on Graphs , 1977 .

[44]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.

[45]  T. C. Hu,et al.  Multi-Terminal Network Flows , 1961 .

[46]  R. Kipp Martin,et al.  Using separation algorithms to generate mixed integer model reformulations , 1991, Oper. Res. Lett..

[47]  Martin Grötschel,et al.  The ellipsoid method and its consequences in combinatorial optimization , 1981, Comb..

[48]  Giovanni Rinaldi,et al.  A Branch-and-Cut Algorithm for the Resolution of Large-Scale Symmetric Traveling Salesman Problems , 1991, SIAM Rev..

[49]  Hendrik W. Lenstra,et al.  Integer programming and cryptography , 1984 .

[50]  Martin W. P. Savelsbergh,et al.  MINTO, a mixed INTeger optimizer , 1994, Oper. Res. Lett..

[51]  C. A. Rogers,et al.  An Introduction to the Geometry of Numbers , 1959 .

[52]  Zoltán Füredi,et al.  Computing the volume is difficult , 1986, STOC '86.

[53]  C. McDiarmid.  Rado's theorem for polymatroids , 1975, Mathematical Proceedings of the Cambridge Philosophical Society.

[54]  N. Z. Shor.  Convergence rate of the gradient descent method with dilatation of the space , 1970 .

[55]  Ellis L. Johnson,et al.  Solving Large-Scale Zero-One Linear Programming Problems , 1983, Oper. Res..

[56]  Ravi Kannan,et al.  Minkowski's Convex Body Theorem and Integer Programming , 1987, Math. Oper. Res..

[57]  Alexander Schrijver,et al.  Cones of Matrices and Set-Functions and 0-1 Optimization , 1991, SIAM J. Optim..

[58]  J. K. Lowe.  Modelling with Integer Variables. , 1984 .

[59]  J. F. Benders.  Partitioning procedures for solving mixed-variables programming problems , 1962 .

[60]  M. R. Rao,et al.  Solving the Steiner Tree Problem on a Graph Using Branch and Cut , 1992, INFORMS J. Comput..

[61]  D. George Wilson,et al.  Introduction to the IBM Optimization Subroutine Library , 1992, IBM Syst. J..

[62]  Michele Conforti,et al.  Testing balancedness and perfection of linear matrices , 1993, Math. Program..

[63]  William H. Cunningham,et al.  Testing membership in matroid polyhedra , 1984, J. Comb. Theory, Ser. B.

[64]  Leslie E. Trotter,et al.  Properties of vertex packing and independence system polyhedra , 1974, Math. Program..

[65]  Jack Edmonds,et al.  Maximum matching and a polyhedron with 0,1-vertices , 1965 .

[66]  Ravindra K. Ahuja,et al.  Network Flows: Theory, Algorithms, and Applications , 1993 .

[67]  H. P. Williams.  Linear and integer programming applied to the propositional calculus , 1987 .

[68]  J. Hooker,et al.  Logic-Based Methods for Optimization: Combining Optimization and Constraint Satisfaction , 2000 .

[69]  Vasek Chvátal,et al.  Edmonds polytopes and a hierarchy of combinatorial problems , 1973, Discret. Math..

[70]  Jeffrey C. Lagarias,et al.  Worst-Case Complexity Bounds for Algorithms in the Theory of Integral Quadratic Forms , 1980, J. Algorithms.

[71]  R E Gomory,et al.  ON THE RELATION BETWEEN INTEGER AND NONINTEGER SOLUTIONS TO LINEAR PROGRAMS. , 1965, Proceedings of the National Academy of Sciences of the United States of America.

[72]  Richard M. Karp,et al.  The Traveling-Salesman Problem and Minimum Spanning Trees , 1970, Oper. Res..

[73]  László Lovász,et al.  On the Shannon capacity of a graph , 1979, IEEE Trans. Inf. Theory.

[74]  D. R. Fulkerson,et al.  On balanced matrices , 1974 .

[75]  Eugene L. Lawler,et al.  Matroid intersection algorithms , 1975, Math. Program..

[76]  Jack Edmonds,et al.  Matroids and the greedy algorithm , 1971, Math. Program..

[77]  Carsten Lund,et al.  Proof verification and hardness of approximation problems , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[78]  Pascal Van Hentenryck.  Principles and Practice of Constraint Programming - CP96. Second International Conference, CP96, Cambridge, MA, USA, August 19-22, 1996, Proceedings. , 1997 .

[79]  Manfred W. Padberg,et al.  Lehman's forbidden minor characterization of ideal 0-1 matrices , 1993, Discret. Math..

[80]  J. Shapiro.  A Survey of Lagrangian Techniques for Discrete Optimization. , 1979 .

[81]  Gérard Cornuéjols,et al.  A class of logic problems solvable by linear programming , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[82]  Sartaj Sahni,et al.  Computationally Related Problems , 1974, SIAM J. Comput..

[83]  L. Khachiyan.  Polynomial algorithms in linear programming , 1980 .