Skip to main content
SpringerLink
Log in

Deriving compact extended formulations via LP-based separation techniques

  • SI: 4OR Surveys
  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

The best formulations for some combinatorial optimization problems are integer linear programming models with an exponential number of rows and/or columns, which are solved incrementally by generating missing rows and columns only when needed. As an alternative to row generation, some exponential formulations can be rewritten in a compact extended form, which have only a polynomial number of constraints and a polynomial, although larger, number of variables. As an alternative to column generation, there are compact extended formulations for the dual problems, which lead to compact equivalent primal formulations, again with only a polynomial number of constraints and variables. In this this paper we introduce a tool to derive compact extended formulations and survey many combinatorial optimization problems for which it can be applied. The tool is based on the possibility of formulating the separation procedure by an LP model. It can be seen as one further method to generate compact extended formulations besides other tools of geometric and combinatorial nature present in the literature.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  • Appelgren, L. (1969). A column generation algorithm for a ship scheduling problem. Transportation Science, 3, 53–68.

    Article  Google Scholar 

  • Barahona, F. (1993). On cuts and matchings in planar graphs. Mathematical Programming, 60, 53–68.

    Article  Google Scholar 

  • Barahona, F., Jünger, M., & Reinelt, G. (1989). Experiments in quadratic 0–1 programming. Mathematical Programming, 44, 127–137.

    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(3), 316–329.

    Article  Google Scholar 

  • Berman, H. M., Westbrook, J., Feng, Z., Gilliland, G., Bhat, T. N., Weissig, H., et al. (2000). The protein data bank. Nucleic Acids Research, 28, 235–242.

    Article  Google Scholar 

  • Bertsimas, D., & Sim, M. (2003). Robust discrete optimization and network flows. Mathematical Programming, 98, 49–71.

    Article  Google Scholar 

  • Bertsimas, D., & Sim, M. (2004). The price of robustness. Operations Research, 52, 35–53.

    Article  Google Scholar 

  • Caprara, A. (1999). Sorting permutations by reversals and Eulerian cycle decompositions. SIAM Journal on Discrete Mathematics, 12, 91–110.

    Article  Google Scholar 

  • Caprara, A., Carr, R. D., Lancia, G., Walenz, B., & Istrail, S. (2004). 1001 optimal PDB structure alignments: Integer programming methods for finding the maximum contact map overlap. Journal of Computational Biology, 11, 27–52.

    Article  Google Scholar 

  • Caprara, A., Lancia, G., & Ng, S. K. (2001). Sorting permutations by reversal through branch-and-price. Informs Journal on Computing, 13, 224–244.

    Article  Google Scholar 

  • Caprara, A., Panconesi, A., & Rizzi, R. (2003). Packing cycles in undirected graphs. Journal of Algorithms, 48, 239–256.

    Article  Google Scholar 

  • Carr, R. D., & Lancia, G. (2002). Compact vs. exponential-size LP relaxations. Operations Research Letters, 30, 57–65.

    Article  Google Scholar 

  • Carr, R. D., & Lancia, G. (2004). Compact optimization can outperform separation: A case study in structural proteomics. 4OR, 2, 221–233.

    Article  Google Scholar 

  • Chvátal, V. (1975). On certain polytopes associated with graphs. Journal of Combinatorial Theory Series B, 18, 138–154.

    Article  Google Scholar 

  • Conforti, M., Cornuéjols, G., & Zambelli, G. (2010). Extended formulations in combinatorial optimization. 4OR, 8, 1–48.

    Article  Google Scholar 

  • Cook, W. J., Cunningham, W. H., Pulleyblank, W. R., & Schrijver, A. (1998). Combinatorial optimization. New York: Wiley.

    Google Scholar 

  • Dantzig, G. B., Fulkerson, R., & Johnson, S. M. (1954). Solution of a large-scale traveling salesman problem. Operations Research, 2, 393–410.

    Google Scholar 

  • de Carvalho, J. M. V. (1999). Exact solutions of bin-packing problems using column generation and branch-and-bound. Annals of Operations Research, 86, 629–665.

    Article  Google Scholar 

  • de Carvalho, J. M. V. (2002). LP models for bin packing and cutting stock problems. European Journal of Operational Research, 141, 253–273.

    Article  Google Scholar 

  • De Simone, C., & Rinaldi, G. (1994). A cutting plane algorithm for the max-cut problem. Optimization Methods and Software, 3, 195–214.

    Article  Google Scholar 

  • Fiorini, S., Massar, S., Pokutta, S., Raj Tiwary, H., & de Wolf, R. (2012). Linear vs. semidefinite extended formulations: Exponential separation and strong lower bounds. In 44th ACM symposium on theory of computing (STOC 2012), New York, NY, USA, 19–22 May 2012.

  • Fischetti, M., Lancia, G., & Serafini, P. (2002). Exact algorithms for minimum routing cost trees. Networks, 39, 1–13.

    Article  Google Scholar 

  • Fischetti, M., & Monaci, M. (2012). Cutting plane versus compact formulations for uncertain (integer) linear programs. Mathematical Programming Computation, 4, 239–273.

    Article  Google Scholar 

  • Gerards, A. M. H., & Schrijver, A. (1986). Matrices with the Edmonds–Johnson property. Combinatorica, 6, 365–379.

    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 

  • Gilmore, P. C., & Gomory, R. E. (1963). A linear programming approach to the cutting stock problem—II. Operations Research, 11, 863–888.

    Article  Google Scholar 

  • Goldman, D., Istrail, S., & Papadimitriou, C. (1999). Algorithmic aspects of protein structure similarity. In Proceedings of the 40th annual IEEE symposium on foundations of computer science (pp. 512–522).

  • Grötschel, M., Jünger, M., & Reinelt, G. (1987). Calculating exact ground states of spin glasses: A polyhedral approach. In: J.L. van Hemmen & I. Morgenstern (Eds.), Heidelberg colloquium on glassy dynamics (pp. 325–353). Berlin: Springer.

  • Grötschel, M., & Holland, O. (1991). Solution of large-scale travelling salesman problems. Mathematical Programming, 51(2), 141–202.

    Article  Google Scholar 

  • Grötschel, M., Lovász, L., & Schrijver, A. (1981). The ellipsoid method and its consequences in combinatorial optimization. Combinatorica, 1, 169–197.

    Article  Google Scholar 

  • Grötschel, M., Lovász, L., & Schrijver, A. (1988). Geometric algorithms and combinatorial optimization. Berlin: Springer.

    Book  Google Scholar 

  • Hu, T. C. (1974). Optimum communication spanning trees. SIAM Journal on Computing, 3, 188–195.

    Article  Google Scholar 

  • Kaibel, V. (2011). Extended formulations in combinatorial optimization. arXiv preprint arXiv:1104.1023.

  • Kaibel, V., & Pashkovich, K. (2011). Constructing extended formulations from reflection relations. In: O. Günlük & G. Woeginger (Eds.), Integer programming and combinatorial optimization XV, lecture notes in computer science 6655 (pp. 287–300). Springer.

  • Lancia, G., Carr, R. D., Walenz, B., & Istrail, S. (2001). 101 optimal PDB structure alignments: a branch-and-cut algorithm for the maximum contact map overlap problem. In: Proceedings of 5th ACM international conference on computational molecular biology (RECOMB) (pp. 193–202).

  • Lancia, G., Rinaldi, F., & Serafini, P. (2011). A time-indexed LP-based approach for min-sum job-shop problems. Annals of Operations Research, 86, 175–198.

    Article  Google Scholar 

  • Lancia, G., & Serafini, P. (2011). An effective compact formulation of the max cut problem on sparse graphs. Electronic Notes in Discrete Mathematics, 37, 111–116.

    Article  Google Scholar 

  • Lancia, G., & Serafini, P. (2014). Deriving compact extended formulations via LP-based separation techniques. 4OR-A Quarterly Journal of Operations Research, 12, 201–234.

    Article  Google Scholar 

  • Lenhof, H. P., Reinert, K., & Vingron, M. (1998). A polyhedral approach to RNA sequence structure alignment. Journal of Computational Biology, 5, 517–530.

    Article  Google Scholar 

  • Martin, K. (1991). Using separation algorithms to generate mixed integer model reformulations. Operations Research Letters, 10, 119–128.

    Article  Google Scholar 

  • Monaci, M., & Pferschy, U. (2011). On the robust knapsack problem. SIAM Journal on Optimization, 23, 1956–1982.

    Article  Google Scholar 

  • Monaci, M., Pferschy, U., & Serafini, P. (2013). Exact solution of the robust knapsack problem. Computers & Operations Research, 40, 2625–2631.

    Article  Google Scholar 

  • Newman, A. (2008). Max cut. In M.-Y. Kao (Ed.), Encyclopedia of algorithms (pp. 1–99). New York: Springer.

    Google Scholar 

  • Padberg, M., & Rinaldi, G. (1991). A branch-and-cut algorithm for the resolution of large-scale symmetric traveling salesman problems. SIAM Review, 33(1), 60–100.

    Article  Google Scholar 

  • Stoer, M., & Wagner, F. (1994). A simple mincut algorithm. In Proceedings of ESA 94, lecture notes in computer science 855 (pp. 141–147). Berlin: Springer.

  • Villeneuve, D., Desrosiers, J., Lübbecke, M. E., & Soumis, F. (2005). On compact formulations for integer programs solved by column generation. Annals of Operations Research, 139, 375–388.

    Article  Google Scholar 

  • Wu, B. Y., Lancia, G., Bafna, V., Chao, K. M., Ravi, R., & Tang, C. Y. (1999). A polynomial-time approximation scheme for minimum routing cost spanning trees. SIAM Journal on Computing, 29, 761–778.

    Article  Google Scholar 

  • Yannakakis, M. (1991). Expressing combinatorial optimization problems by linear programs. Journal of Computer and System Sciences, 43, 441–466.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, University of Udine, Via delle Scienze 206, 33100, Udine, Italy

    Giuseppe Lancia & Paolo Serafini

Authors
  1. Giuseppe Lancia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paolo Serafini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paolo Serafini.

Additional information

This is an updated version of the paper that appeared in 4OR, 12(3), 201–234, (2014).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lancia, G., Serafini, P. Deriving compact extended formulations via LP-based separation techniques. Ann Oper Res 240, 321–350 (2016). https://doi.org/10.1007/s10479-015-2012-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-015-2012-4

Keywords

Search

Navigation