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Algorithms for Inverse Optimization Problems

Authors Sara Ahmadian, Umang Bhaskar, Laura Sanità, Chaitanya Swamy

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Sara Ahmadian
  • Combinatorics and Optimization, Univ. Waterloo, Waterloo, ON N2L 3G1, Canada
Umang Bhaskar
  • Tata Institute of Fundamental Research, Mumbai, India 400 005
Laura Sanità
  • Combinatorics and Optimization, Univ. Waterloo, Waterloo, ON N2L 3G1, Canada
Chaitanya Swamy
  • Combinatorics and Optimization, Univ. Waterloo, Waterloo, ON N2L 3G1, Canada

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Sara Ahmadian, Umang Bhaskar, Laura Sanità, and Chaitanya Swamy. Algorithms for Inverse Optimization Problems. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 1:1-1:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


We study inverse optimization problems, wherein the goal is to map given solutions to an underlying optimization problem to a cost vector for which the given solutions are the (unique) optimal solutions. Inverse optimization problems find diverse applications and have been widely studied. A prominent problem in this field is the inverse shortest path (ISP) problem [D. Burton and Ph.L. Toint, 1992; W. Ben-Ameur and E. Gourdin, 2004; A. Bley, 2007], which finds applications in shortest-path routing protocols used in telecommunications. Here we seek a cost vector that is positive, integral, induces a set of given paths as the unique shortest paths, and has minimum l_infty norm. Despite being extensively studied, very few algorithmic results are known for inverse optimization problems involving integrality constraints on the desired cost vector whose norm has to be minimized. Motivated by ISP, we initiate a systematic study of such integral inverse optimization problems from the perspective of designing polynomial time approximation algorithms. For ISP, our main result is an additive 1-approximation algorithm for multicommodity ISP with node-disjoint commodities, which we show is tight assuming P!=NP. We then consider the integral-cost inverse versions of various other fundamental combinatorial optimization problems, including min-cost flow, max/min-cost bipartite matching, and max/min-cost basis in a matroid, and obtain tight or nearly-tight approximation guarantees for these. Our guarantees for the first two problems are based on results for a broad generalization, namely integral inverse polyhedral optimization, for which we also give approximation guarantees. Our techniques also give similar results for variants, including l_p-norm minimization of the integral cost vector, and distance-minimization from an initial cost vector.

Subject Classification

ACM Subject Classification
  • Theory of computation → Network optimization
  • Theory of computation → Approximation algorithms analysis
  • Mathematics of computing → Network flows
  • Inverse optimization
  • Shortest paths
  • Approximation algorithms
  • Linear programming
  • Polyhedral theory
  • Combinatorial optimization


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  1. Ravindra K Ahuja, Thomas L Magnanti, and James B Orlin. Network flows: theory, algorithms, and applications. Prentice hall, 1993. Google Scholar
  2. R.K. Ahuja and J.B. Orlin. Inverse optimization. Oper. Res., 49:171-783, 2001. Google Scholar
  3. W. Ben-Ameur and E. Gourdin. Internet routing and related topology issues. SIAM J. Discrete Math., 17:18-49, 2004. Google Scholar
  4. A. Bley. Inapproximability results for the inverse shortest paths problem with integer lengths and unique shortest paths. Networks, 50:29-36, 2007. Google Scholar
  5. Endre Boros, Khaled Elbassioni, Vladimir Gurvich, and Hans Raj Tiwary. The negative cycles polyhedron and hardness of checking some polyhedral properties. Annals of Operations Research, 188(1):63-76, 2011. Google Scholar
  6. David Bremner, Komei Fukuda, and Ambros Marzetta. Primal-dual methods for vertex and facet enumeration. Discrete &Computational Geometry, 20(3):333-357, 1998. Google Scholar
  7. D. Burton, W. Pulleyblank, and Ph. L. Toint. The inverse shortest paths problem with upper bounds on shortest paths costs. In Network optimization, pages 156-171. Springer, 1996. Google Scholar
  8. D Burton and Ph L Toint. On the use of an inverse shortest paths algorithm for recovering linearly correlated costs. Mathematical Programming, 63(1):1-22, 1994. Google Scholar
  9. D. Burton and Ph.L. Toint. On an instance of the inverse shortest paths problem. Math. Program., 53:45-61, 1992. Google Scholar
  10. Michael R Bussieck and Marco E Lübbecke. The vertex set of a 01-polytope is strongly p-enumerable. Computational Geometry, 11(2):103-109, 1998. Google Scholar
  11. Mikael Call and Kaj Holmberg. Complexity of inverse shortest path routing. In INOC, pages 339-353. Springer, 2011. Google Scholar
  12. Robert B Dial. Minimal-revenue congestion pricing part i: A fast algorithm for the single-origin case. Transportation Research Part B: Methodological, 33(3):189-202, 1999. Google Scholar
  13. Robert B Dial. Minimal-revenue congestion pricing part ii: An efficient algorithm for the general case. Transportation Research Part B: Methodological, 34(8):645-665, 2000. Google Scholar
  14. Martin E Dyer. The complexity of vertex enumeration methods. Mathematics of Operations Research, 8(3):381-402, 1983. Google Scholar
  15. F. Grandoni, G. Nicosia, G. Oriolo, and L. Sanità. Stable routing under the spanning tree protocol. Operations Research Letters, 38:399-404, 2010. Google Scholar
  16. N. Haehnle, L. Sanità, and R. Zenklusen. Stable routing and unique-max coloring on trees. SIAM J. Discret. Math, 27:109-125, 2013. Google Scholar
  17. C. Heuberger. Inverse combinatorial optimization: A survey on problems, methods, and results. J. Combin. Optim., 8:329-361, 2004. Google Scholar
  18. Garud Iyengar and Wanmo Kang. Inverse conic programming with applications. Operations Research Letters, 33(3):319-330, 2005. Google Scholar
  19. Gertrud Neumann-Denzau and Jörn Behrens. Inversion of seismic data using tomographical reconstruction techniques for investigations of laterally inhomogeneous media. Geophysical Journal International, 79(1):305-315, 1984. Google Scholar
  20. J. Scott Provan. Efficient enumeration of the vertices of polyhedra associated with network LP’s. Math. Program., 63(1):47-64, 1994. Google Scholar
  21. Robert C Read. Bounds on backtrack algorithms for listing cycles, paths and spanning trees. Networks, 5:237-252, 1975. Google Scholar
  22. Alexander Schrijver. Theory of linear and integer programming. John Wiley &Sons, 1998. Google Scholar
  23. Albert Tarantola. Inverse problem theory and methods for model parameter estimation. SIAM, 2005. Google Scholar
  24. H.R. Varian. Revealed preference. Samuelsonian Economics and the Twenty-first Century, pages 99-115, 2006. Google Scholar
  25. S. Xu and J. Zhang. An inverse problem of the weighted shortest path problem. Japan J. Indust. Appl. Math., 12:47-59, 1995. Google Scholar
  26. J. Zhang and M.C. Cai. Inverse problem of minimum cuts. Mathematical Methods of Oper. Res., 47:51-58, 1998. Google Scholar
  27. J. Zhang and Z. Liu. A general model of some inverse combinatorial optimization problems and its solution method under l-infinity norm. J. Combin. Optim., 6:207-227, 2002. Google Scholar
  28. J. Zhang and Z. Ma. Solution structure of some inverse combinatorial optimization problems. J. Combin. Optim., 3:127-139, 1999. Google Scholar
  29. Jianzhong Zhang and Zhenhong Liu. Calculating some inverse linear programming problems. Journal of Computational and Applied Mathematics, 72(2):261-273, 1996. Google Scholar
  30. Jianzhong Zhang and Zhenhong Liu. A further study of inverse linear programming problems. Journal of Computational and Applied Mathematics, 106:345-359, 1999. Google Scholar
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