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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