Algorithms for Inverse Optimization Problems
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.
Inverse optimization
Shortest paths
Approximation algorithms
Linear programming
Polyhedral theory
Combinatorial optimization
Theory of computation~Network optimization
Theory of computation~Approximation algorithms analysis
Mathematics of computing~Network flows
1:1-1:14
Regular Paper
Sara
Ahmadian
Sara Ahmadian
Combinatorics and Optimization, Univ. Waterloo, Waterloo, ON N2L 3G1, Canada
Umang
Bhaskar
Umang Bhaskar
Tata Institute of Fundamental Research, Mumbai, India 400 005
Funded in part by a Ramanujan Fellowship. Part of this work was done while visiting U. Waterloo.
Laura
Sanità
Laura Sanità
Combinatorics and Optimization, Univ. Waterloo, Waterloo, ON N2L 3G1, Canada
Chaitanya
Swamy
Chaitanya Swamy
Combinatorics and Optimization, Univ. Waterloo, Waterloo, ON N2L 3G1, Canada
Supported in part by NSERC grant 327620-09 and an NSERC Discovery Accelerator Supplement Award.
10.4230/LIPIcs.ESA.2018.1
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Sara Ahmadian, Umang Bhaskar, Laura Sanità, and Chaitanya Swamy
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