Skip to main content

Integer and Combinatorial Optimization

  • Reference work entry
  • First Online:
Book cover Encyclopedia of Operations Research and Management Science

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 799.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 899.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Aardal, K., & van Hoesel, C. (1996a). Polyhedral techniques in combinatorial optimization I: Applications and computations. Statistica Neerlandica, 50, 3–26.

    Article  Google Scholar 

  • Aardal, K., & van Hoesel, C. (1996b). Polyhedral techniques in combinatorial optimization II: Applications and computations. Statistica Neerlandica, 50, 3–26.

    Article  Google Scholar 

  • Abara, J. (1989). Applying integer linear programming to the fleet assignment problem. Interfaces, 19, 20–28.

    Article  Google Scholar 

  • Achtenberg, T., & Berthold, T. (2007). Improving the feasibility pump. Discrete Mathematics, 4, 77–86.

    Google Scholar 

  • Achterberg, T., Koch, T., & Martin, A. (2005). Branching rules revisited. Operations Research Letters, 33, 42–54.

    Article  Google Scholar 

  • Ahuja, R., Magnanti, T., & Orlin, J. (1993). Network flows: Theory, algorithms, and applications. Englewood Cliffs, NJ: Prentice Hall.

    Google Scholar 

  • Anderson, E., & Anderson, K. (1995). Presolving in linear programming. Mathematical Programming, 71, 221–245.

    Google Scholar 

  • Applegate, D., & Cook, W. (1991). A computational study of the job-shop scheduling problem. INFORMS Journal on Computing, 3, 149–156.

    Article  Google Scholar 

  • Applegate, D., Bixby, R., Chvátal, V., & Cook, W. (2006). The traveling salesman problem: A computational study. Princeton, NJ: Princeton University Press.

    Google Scholar 

  • Atamturk, A., & Savelsbergh, M. (2000). Conflict graphs in solving integer programming problems. European Journal of Operational Research, 121, 40–55.

    Article  Google Scholar 

  • Atamtürk, A., & Savelsbergh, M. (2005). Integer-programming software systems. Annals of Operations Research, 140, 67–124.

    Article  Google Scholar 

  • Avner, P. (2001). A radiation hybrid transcript may of the mouse genome. Nature Genetics, 29, 194–200.

    Article  Google Scholar 

  • Balas, E. (1975). Facets of the knapsack polytope. Mathematical Programming, 8, 146–164.

    Article  Google Scholar 

  • Balas, E. (1998). Disjunctive programming: Properties of the convex hull of feasible points. Discrete Applied Mathematics, 89, 3–44.

    Article  Google Scholar 

  • Balas, E., & Martin, R. (1980). Pivot and complement: A heuristic for 0-1 programming. Management Science, 26, 86–96.

    Article  Google Scholar 

  • Balas, E., & Padberg, M. (1972). On the set-covering problem. Operations Research, 20, 1152–1161.

    Article  Google Scholar 

  • Balas, E., & Padberg, M. (1976). Set partitioning: A survey. SIAM Review, 18, 710–760.

    Article  Google Scholar 

  • Balas, E., Ceria, S., & Corneujols, G. (1993). A lift-and-project cutting plane algorithm for mixed 0-1 programs. Mathematical Programming, 58, 295–324.

    Article  Google Scholar 

  • Balas, E., Ceria, S., & Cornuejols, G. (1996). Mixed 0-1 programming by lift-and-project in a branch-and-cut framework. Management Science, 42, 1229–1246.

    Article  Google Scholar 

  • Balas, E., Ceria, S., Cornuejols, G., & Natraj, N. (1999). Gomory cuts revisited. Operations Research Letters, 19, 1–9.

    Article  Google Scholar 

  • Balas, E., Schmieta, S., & Wallace, C. (2004). Pivot and shift—A mixed integer programming heuristic. Discrete Optimization, 1, 3–12.

    Article  Google Scholar 

  • Barahona, F., Grötschel, M., Jünger, M., & Reinelt, G. (1988). An application of combinatorial optimization to statistical physics and circuit layout design. Operations Research, 36, 493–513.

    Article  Google Scholar 

  • Barnhart, C., Johnson, E. L., Nemhauser, G. L., Savelsbergh, M. W. P., & Vance, P. H. (1998). Branch and price: Column generation for solving huge integer programs. Operations Research, 46, 316–329.

    Article  Google Scholar 

  • Bauer, P., Linderoth, J., & Savelsbergh, M. (2002). A branch and cut approach to the cardinality constrained circuit problem. Mathematical Programming, 9, 307–348.

    Article  Google Scholar 

  • Benders, J. F. (1962). Partitioning procedures for solving mixed variable programming problems. Numerische Mathematik, 4, 238–252.

    Article  Google Scholar 

  • Bertsimas, D., & Weismantel, R. (2005). Optimization over integers. Cambridge, MA: Dynamic Ideas.

    Google Scholar 

  • Borndörfer, R., & Weismantel, R. (2000). Set packing relaxations of some integer programs. Mathematical Programming, 88, 425–450.

    Article  Google Scholar 

  • Bramel, J., & Simchi-Levi, D. (1997). On the effectiveness of set covering formulations for the vehicle routing problem with time windows. Operations Research, 45, 295–301.

    Article  Google Scholar 

  • Brearley, A., Mitra, G., & Williams, H. (1975). Analysis of mathematical programming problems prior to applying the simplex method. Mathematical Programming, 8, 54–83.

    Article  Google Scholar 

  • Brooke, A., Kendrick, D., & Meeraus, A. (1988). GAMS, a user’s guide. Redwood City, CA: The Scientific Press.

    Google Scholar 

  • Chabrier, A. (2006). Vehicle routing problem with elementary shortest path based column generation. Computers and Operations Research, 33(10), 2972–2990.

    Article  Google Scholar 

  • Chan, L., Muriel, A., & Simchi-Levi, D. (1998a). Parallel machine scheduling, linear programming, and parameter list scheduling heuristics. Operations Research, 46, 729–741.

    Article  Google Scholar 

  • Chan, L., Simchi-Levi, D., & Bramel, J. (1998b). Worst-case analyses, linear programming and the bin-packing problem. Mathematical Programming, 83, 213–227.

    Google Scholar 

  • Chvátal, V. (1983). Linear programming. New York: W. H. Freeman.

    Google Scholar 

  • Cornuéjols, G. (2008). Valid inequalities for mixed integer linear programs. Mathematical Programming B, 112, 3–44.

    Article  Google Scholar 

  • Cornuejols, G., Nemhauser, G., & Wolsey, L. (1980). Worst-case and probabilistic analysis of algorithms for a location problem. Operations Research, 28, 847–858.

    Article  Google Scholar 

  • Cramton, P., Shoham, Y., & Steinberg, R. (2006). Combinatorial auctions. Cambridge, MA: MIT Press.

    Google Scholar 

  • Danna, E., Rothberg, E., & LePape, C. (2005). Exploring relaxation induced neighborhoods to improve MIP solutions. Mathematical Programming, 102, 71–90.

    Article  Google Scholar 

  • Dantzig, G., & Wolfe, P. (1960). Decomposition principle for linear programs. Operations Research, 8, 101–111.

    Article  Google Scholar 

  • Drezner, Z., & Hamacher, H. (2004). Facility location: Applications and theory. Berlin: Springer.

    Google Scholar 

  • Edmonds, J., & Johnson, E. L. (1973). Matching, Euler tours, and the Chinese postman. Mathematical Programming, 5, 88–124.

    Article  Google Scholar 

  • Fischetti, M., & Lodi, A. (2002). Local branching. Mathematical Programming, 98, 23–47.

    Article  Google Scholar 

  • Fischetti, M., Lodi, A., & Salvagnin, D. (2009). Just MIP It! In V. Maniezzo, T. Stuetzle, & S. Voss (Eds.), MATHEURISTICS: Hybridizing metaheuristics and mathematical programming (pp. 39–70). Berlin: Springer.

    Chapter  Google Scholar 

  • Fisher, M. L. (1981). The lagrangian method for solving integer programming problems. Management Science, 27, 1–18.

    Article  Google Scholar 

  • Fourer, R., Gay, D. M., & Kernighan, B. W. (1993). AMPL: A modeling language for mathematical programming. San Francisco: The Scientific Press.

    Google Scholar 

  • Garey, M. R., & Johnson, D. S. (1979). Computers and intractability: A guide to the theory of NP-completeness. New York: W. H. Freeman.

    Google Scholar 

  • Geoffrion, A. (1974). Lagrangian relaxation for integer programming. Mathematical Programming Study, 2, 82–114.

    Article  Google Scholar 

  • Gilmore, P. C., & Gomory, R. E. (1961). A linear programming approach to the cutting stock problem. Operations Research, 9, 849–859.

    Article  Google Scholar 

  • Glover, F., & Laguna, M. (1998). Tabu search. Berlin: Springer.

    Book  Google Scholar 

  • Goldberg, D. (1989). Genetic algorithms in search, optimization, and machine learning. Reading, MA: Addison-Wesley.

    Google Scholar 

  • Golden, B., Raghavan, S., & Wasail, E. (2010). The vehicle routing problem: Latest advances and new challenges. Berlin: Springer.

    Google Scholar 

  • Gomory, R. E. (1958). Outline of an algorithm for integer solutions to linear programs. Bulletin of the American Mathematical Monthly, 64, 275–278.

    Article  Google Scholar 

  • Gomory, R. E. (1960). An algorithm for the mixed integer problem (Tech. Rep. RM-2597). The RAND Corporation. Santa Monica, California.

    Google Scholar 

  • Gu, Z., Nemhauser, G. L., & Savelsbergh, M. W. P. (1998). Cover inequalities for 0-1 linear programs: Computation. INFORMS Journal on Computing, 10, 427–437.

    Article  Google Scholar 

  • Gu, Z., Nemhauser, G. L., & Savelsbergh, M. W. P. (1999). Lifted flow covers for mixed 0-1 integer programs. Mathematical Programming, 85, 439–467.

    Article  Google Scholar 

  • Gu, Z., Nemhauser, G. L., & Savelsbergh, M. W. P. (2000). Sequence independent lifting. Journal of Combinatorial Optimization, 4, 109–129.

    Article  Google Scholar 

  • Guignard, M., & Kim, S. (1987). Lagrangian decomposition: A model yielding stronger lagrangian bounds. Mathematical Programming, 39, 215–228.

    Article  Google Scholar 

  • Guignard, M., & Spielberg, K. (1981). Logical reduction methods in zero-one programming: Minimal preferred inequalities. Operations Research, 29, 49–74.

    Article  Google Scholar 

  • Hammer, P. L., Johnson, E. L., & Peled, U. N. (1975). Facets of regular 0-1 polytopes. Mathematical Programming, 8, 179–206.

    Article  Google Scholar 

  • Hane, C., Barnhart, C., Johnson, E., Marsten, R., Nemhauser, G., & Sigismondi, G. (1995). The fleet assignment problem: Solving a large-scale integer program. Mathematical Programming, 70, 211–232.

    Google Scholar 

  • Hansen, P. (1986). The steepest ascent mildest descent heuristic for combinatorial programming. Proceedings of Congress on Numerical Methods in Combinatorial Optimization, Italy.

    Google Scholar 

  • Held, M., & Karp, R. M. (1970). The traveling salesman problem and minimum spanning trees. Operations Research, 18, 1138–1162.

    Article  Google Scholar 

  • Hoffman, K., & Padberg, M. (1985). LP-based combinatorial problem solving. Annals of Operations Research, 4, 145–194.

    Article  Google Scholar 

  • Hoffman, K. L., & Padberg, M. W. (1991). Improving LP-representations of zero-one linear programs for branch and cut. ORSA Journal on Computing, 3, 121–134.

    Article  Google Scholar 

  • Hoffman, K., & Padberg, M. (1993). Solving airline crew scheduling problems by branch-and-cut. Management Science, 39, 667–682.

    Google Scholar 

  • Hoffman, K., & Villa, C. (2007). A column-generation and branch-and-cut approach to the bandwidth-packing problem. Journal of Research of the National Institute of Standards and Technology, 111, 161–185.

    Google Scholar 

  • Hooker, J. (2002). Logic, optimization, and constraint programming. INFORMS Journal on Computing, 14, 295–321.

    Article  Google Scholar 

  • Hooker, J. (2007). Planning and scheduling by logic-based benders decomposition. Operations Research, 55, 588–602.

    Article  Google Scholar 

  • Jünger, M., Liebling, T., Naddef, D., Nemhauser, G., Pulleyblank, W., Reinelt, G. (2010). Fifty years of integer programming: 1958–2008. Berlin: Springer.

    Book  Google Scholar 

  • Kan, A. R. (1986). An introduction to the analysis of approximation algorithms. Discrete Applied Mathematics, 14, 111–134.

    Article  Google Scholar 

  • Karamanov, M., & Cornuéjols, G. (2009). Branching on general disjunctions. Mathematical Programming, 128, 403–406.

    Google Scholar 

  • Karp, R. (1976). Probabilistic analysis of partitioning algorithms for the traveling salesman problem. In J. F. Traub (Ed.), Algorithms and complexity: New directions and recent results (pp. 1–19). New York: Academic.

    Google Scholar 

  • Land, A. H., & Doig, A. G. (1960). An automatic method for solving discrete programming problems. Econometrica, 28, 497–520.

    Article  Google Scholar 

  • Linderoth, J., & Ralphs, T. (2005). Noncommercial software for mixed-integer linear programming. In J. Karlof (Ed.), Integer programming: Theory and practice (pp. 253–303). Boca Raton, FL: CRC Press.

    Google Scholar 

  • Linderoth, J. T., & Savelsbergh, M. W. P. (1999). A computational study of search strategies in mixed integer programming. INFORMS Journal on Computing, 11, 173–187.

    Article  Google Scholar 

  • Marchand, H., & Wolsey, L. (2001). Aggregation and mixed integer rounding to solve MIPs. Operations Research, 49, 363–371.

    Article  Google Scholar 

  • Markowitz, H., & Manne, A. (1957). On the solution of discrete programming problems. Econometrica, 2, 84–110.

    Article  Google Scholar 

  • Martello, S., & Toth, P. (1990). Knapsack problems. New York: Wiley.

    Google Scholar 

  • Martin, A. (2001). Computational issues for branch-and-cut algorithms. In M. Juenger & D. Naddef (Eds.), Computational combinatorial optimization (pp. 1–25). Berlin: Springer.

    Chapter  Google Scholar 

  • McAloon, K., & Tretkoff, C. (1996). Optimization and computational logic. New York: Wiley.

    Google Scholar 

  • Mitten, L. (1970). Branch-and-bound methods: General formulation and properties. Operations Research, 18, 24–34.

    Article  Google Scholar 

  • Nediak, M., & Eckstein, J. (2001). Pivot, cut, and dive: A heuristic for mixed 0-1 integer programming (Tech. Rep. RUTCOR Research Report RRR 53-2001). Rutgers University, Newark, New Jersey.

    Google Scholar 

  • Nemhauser, G. L., & Sigismondi, G. (1992). A strong cutting plane/branch-and-bound algorithm for node packing. Journal of the Operational Research Society, 43, 443–457.

    Article  Google Scholar 

  • Nemhauser, G. L., & Trotter, L. E., Jr. (1974). Properties of vertex packing and independence system polyhedra. Mathematical Programming, 6, 48–61.

    Article  Google Scholar 

  • Nemhauser, G., & Vance, P. (1994). Lifted cover facets of the 0-1 knapsack polytope with GUB constraints. Operations Research Letters, 16, 255–264.

    Article  Google Scholar 

  • Nemhauser, G., & Wolsey, L. A. (1988). Integer and combinatorial optimization. New York: Wiley.

    Book  Google Scholar 

  • Nemhauser, G., & Wolsey, L. (1990). A recursive procedure for generating all cuts for 0-1 mixed integer programs. Mathematical Programming, 46, 379–390.

    Article  Google Scholar 

  • Odlyzko, A. M. (1990). The rise and fall of knapsack cryptosystems. In C. Pomerance (Ed.), Cryptology and computational number theory (pp. 75–88). Ann Arbor: American Mathematical Society.

    Google Scholar 

  • Owen, J., & Mehrotra, S. (2001). Experimental results on using general disjunctions in branch-and-bound for general-integer linear programs. Computational Optimization and Applications, 20(2).

    Google Scholar 

  • Padberg, M. (1973). On the facial structure of set packing polyhedra. Mathematical Programming, 5, 199–215.

    Article  Google Scholar 

  • Padberg, M. (1974). Perfect zero-one matrices. Mathematical Programming, 6, 180–196.

    Article  Google Scholar 

  • Padberg, M. (1979a). Covering, packing and knapsack problems. Annals of Discrete Mathematics, 4, 265–287.

    Article  Google Scholar 

  • Padberg, M. W. (1979b). A note on 0-1 programming. Operations Research, 23, 833–837.

    Article  Google Scholar 

  • Padberg, M. W., & Rinaldi, G. (1991). A branch and cut algorithm for the solution of large scale traveling salesman problems. SIAM Review, 33, 60–100.

    Article  Google Scholar 

  • Papadimitriou, C., & Steiglitz, K. (1982). Combinatorial optimization: Algorithms and complexity. New Jersey: Prentice-Hall.

    Google Scholar 

  • Parker, R., & Rardin, R. (1988). Discrete optimization. San Diego: Academic.

    Google Scholar 

  • Parker, M., & Ryan, J. (1995). A column generation algorithm for bandwidth packing. Telecommunication Systems, 2, 185–196.

    Article  Google Scholar 

  • Pinedo, M. (2008). Scheduling: Theory, algorithms, and systems. Berlin: Springer.

    Google Scholar 

  • Pochet, Y., & Wolsey, L. (1991). Solving multi-item lot sizing problems using strong cutting planes. Management Science, 37, 53–67.

    Article  Google Scholar 

  • Ralphs, T., & Galati, M. (2005). Decomposition in integer programming. In J. Karlof (Ed.), Integer programming: Theory and practice (pp. 57–110). Boca Raton, FL: CRC Press.

    Google Scholar 

  • Rasmussen, R., & Trick, M. (2007). A benders approach to the constrained minimum break problem. European Journal of Operational Research, 177, 198–213.

    Article  Google Scholar 

  • Ravikumar, C. (1996). Parallel methods for VLSI layout design. Norwood, NJ: Ablex Publishing Corporation.

    Google Scholar 

  • Rendl, F. (2010). Semidefinite relaxations for integer programming. In M. Jünger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, & L. Wolsey (Eds.), Fifty years of integer programming: 1958–2008 (pp. 687–726). Berlin: Springer.

    Chapter  Google Scholar 

  • Rothberg, E. (2007). An evolutionary algorithm for polishing mixed integer programming solutions. INFORMS Journal on Computing, 19, 534–541.

    Article  Google Scholar 

  • Roy, T. J. V., & Wolsey, L. A. (1987). Solving mixed integer 0-1 programs by automatic reformulation. Operations Research, 35, 45–57.

    Article  Google Scholar 

  • Savelsbergh, M. W. P. (1994). Preprocessing and probing techniques for mixed integer programming problems. ORSA Journal on Computing, 6, 445–454.

    Article  Google Scholar 

  • Savelsbergh, M. W. P. (1997). A branch and price algorithm for the generalized assignment problem. Operations Research, 45, 831–841.

    Article  Google Scholar 

  • Schrijver, A. (1986). Theory of linear and integer programming. Chichester: Wiley.

    Google Scholar 

  • Schrijver, A. (2003). Combinatorial optimization: Polyhedra and efficiency. Berlin: Springer.

    Google Scholar 

  • Shah, R. (1998). Optimization problems in SONET/WDM ring architecture. Master’s Essay, Rutgers University, Newark, NJ.

    Google Scholar 

  • Vance, P. H., Barnhart, C., Johnson, E. L., & Nemhauser, G. L. (1994). Solving binary cutting stock problems by column generation and branch and bound. Computational Optimization and Applications, 3, 111–130.

    Article  Google Scholar 

  • Vance, P., Barnhart, C., Johnson, E., & Nemhauser, G. (1997). Airline crew scheduling: A new formulation and decomposition algorithm. Operations Research, 45, 188–200.

    Article  Google Scholar 

  • Vanderbeck, F. (2000). On Dantzig-Wolfe decomposition in integer programming and ways to perform branching in a branch-and-price algorithm. Operations Research, 48, 111–128.

    Article  Google Scholar 

  • Vanderbeck, F., & Wolsey, L. (2010). Reformulation and decomposition of integer programs. In M. Jünger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, & L. Wolsey (Eds.), Fifty years of integer programming: 1958–2008 (pp. 431–504). Berlin: Springer.

    Chapter  Google Scholar 

  • Weingartner, H. (1963). Mathematical programming and the analysis of capital budgeting problems. Englewood Cliffs, NJ: Prentice Hall.

    Google Scholar 

  • Weyl, H. (1935). Elementare theorie der konvexen polyheder. Commentarii Mathematici Helvetici, 7, 290–306.

    Article  Google Scholar 

  • Williams, H. (1985). Model building in mathematical programming (2nd ed.). New York: Wiley.

    Google Scholar 

  • Wolsey, L. A. (1975). Faces for a linear inequality in 0-1 variables. Mathematical Programming, 8, 165–178.

    Article  Google Scholar 

  • Wolsey, L. A. (1976). Facets and strong valid inequalities for integer programs. Operations Research, 24, 367–372.

    Article  Google Scholar 

  • Wolsey, L. A. (1990). Valid inequalities for mixed integer programs with generalized and variable upper bound constraints. Discrete Applied Mathematics, 25, 251–261.

    Article  Google Scholar 

  • Wolsey, L. A. (1998). Integer programming. New York: Wiley.

    Google Scholar 

  • Zhang, X. (2010). Neural networks in optimization. Berlin: Springer.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Karla L. Hoffman .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this entry

Cite this entry

Hoffman, K.L., Ralphs, T.K. (2013). Integer and Combinatorial Optimization. In: Gass, S.I., Fu, M.C. (eds) Encyclopedia of Operations Research and Management Science. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-1153-7_129

Download citation

  • DOI: https://doi.org/10.1007/978-1-4419-1153-7_129

  • Published:

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4419-1137-7

  • Online ISBN: 978-1-4419-1153-7

  • eBook Packages: Business and Economics

Publish with us

Policies and ethics