Skip to main content
Springer Nature Link
Log in

Deriving compact extended formulations via LP-based separation techniques

  • Invited Survey
  • Published:
4OR 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

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Math Program 60:53–68

    Article  Google Scholar 

  • Barahona F, Jünger M, Reinelt G (1989) Experiments in quadratic 0–1 programming. Math Program 44:127–137

    Article  Google Scholar 

  • Barnhart C, Johnson EL, Nemhauser GL, Savelsbergh MWP, Vance PH (1998) Branch-and-price: column generation for solving huge integer programs. Oper Res 46(3):316–329

    Article  Google Scholar 

  • Bertsimas D, Sim M (2003) Robust discrete optimization and network flows. Math Program 98:49–71

    Article  Google Scholar 

  • Bertsimas D, Sim M (2004) The price of robustness. Oper Res 52:35–53

    Article  Google Scholar 

  • Caprara A (1999) Sorting permutations by reversals and Eulerian cycle decompositions. SIAM J Discret Math 12:91–110

    Article  Google Scholar 

  • Caprara A, Lancia G, Ng SK (2001) Sorting permutations by reversal through branch-and-price. Inf J Comput 13:224–244

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Caprara A, Carr RD, Lancia G, Walenz B, Istrail S (2004) 1001 optimal PDB structure alignments: integer programming methods for finding the maximum contact map overlap. J Comput Biol 11:27–52

    Article  Google Scholar 

  • Carr RD, Lancia G (2002) Compact vs exponential-size LP relaxations. Oper Res Lett 30:57–65

    Article  Google Scholar 

  • Carr RD, 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. J Comb Theory Ser 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 WJ, Cunningham WH, Pulleyblank WR, Schrijver A (1998) Combinatorial optimization. Wiley, New York

    Google Scholar 

  • Dantzig GB, Fulkerson R, Johnson SM (1954) Solution of a large-scale traveling salesman problem. Oper Res 2:393–410

    Google Scholar 

  • de Carvalho JMV (1999) Exact solutions of Bin-Packing problems using column generation and branch-and-bound. Ann Oper Res 86:629–665

    Article  Google Scholar 

  • de Carvalho JMV (2002) LP models for bin packing and cutting stock problems. Eur J Oper Res 141:253–273

    Article  Google Scholar 

  • De Simone C, Rinaldi G (1994) A cutting plane algorithm for the max-cut problem. Optim Methods Softw 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

  • Fischetti M, Monaci M (2012) Cutting plane versus compact formulations for uncertain (integer) linear programs. Math Program Comput 4:239–273

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Gilmore PC, Gomory RE (1961) A linear programming approach to the cutting stock problem. Oper Res 9:849–859

    Article  Google Scholar 

  • Gilmore PC, Gomory RE (1963) A linear programming approach to the cutting stock problem-II. Oper Res 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, Holland O (1991) Solution of large-scale travelling salesman problems. Math Program 51(2):141–202

    Article  Google Scholar 

  • Grötschel M, Jünger M, Reinelt G (1987) Calculating exact ground states of spin glasses: a polyhedral approach Heidelberg Colloquium on Glassy Dynamics. Springer, Berlin

    Google Scholar 

  • Hu TC (1974) Optimum communication spanning trees. SIAM J Comp 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 integer programming and combinatorial optimization XV. In: Günlük O, Woeginger G (eds) Lecture Notes in Computer Science 6655. Springer, Berlin, pp 287–300

  • Lancia G, Serafini P (2011) An effective compact formulation of the Max Cut problem on sparse graphs. Electron Notes Discret Math 37:111–116

    Article  Google Scholar 

  • Lancia G, Carr RD, 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. Ann Oper Res 86:175–198

    Article  Google Scholar 

  • Lenhof HP, Reinert K, Vingron M (1998) A polyhedral approach to RNA sequence structure alignment. J Comput Biol 5:517–530

    Article  Google Scholar 

  • Martin K (1991) Using separation algorithms to generate mixed integer model reformulations. Oper Res Lett 10:119–128

    Article  Google Scholar 

  • Monaci M, Pferschy U (2011) On the robust knapsack problem. SIAM J Optim 23:1956–1982

    Article  Google Scholar 

  • Monaci M, Pferschy U, Serafini P (2013) Exact solution of the Robust Knapsack problem. Comput Oper Res 40:2625–2631

    Article  Google Scholar 

  • Newman A (2008) Max Cut. In: Kao Ming-Yang (ed) Encyclopedia of algorithms. Springer, USA, pp 1–99

    Google Scholar 

  • Padberg M, Rinaldi G (1991) A branch-and-cut algorithm for the resolution of large-scale symmetric traveling salesman problems. SIAM Rev 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, vol 855. Springer, Berlin, pp 141–147

  • Wu BY, Lancia G, Bafna V, Chao KM, Ravi R, Tang CY (1999) A polynomial-time approximation scheme for minimum routing cost spanning trees. SIAM J Comp 29:761–778

    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 inPubMed Google Scholar

  2. Paolo Serafini
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Paolo Serafini.

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. 4OR-Q J Oper Res 12, 201–234 (2014). https://doi.org/10.1007/s10288-014-0262-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10288-014-0262-7

Keywords

Mathematics Subject Classification