Optimum turn-restricted paths, nested compatibility, and optimum convex polygons
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We consider two apparently unrelated classes of combinatorial and geometric optimization problems. First, we give compact extended formulations, i.e., polynomial-size linear programming formulations with integer optima, for optimum path problems with turn restrictions satisfying a nested compatibility condition in acyclic digraphs. We then apply these results to optimum convex polygon problems in the plane, by interpreting certain dynamic programming algorithms as sequences of optimum turn-restricted path problems with nested compatibility in acyclic digraphs. As a result, we derive compact extended formulations for these geometric problems as well.
KeywordsShortest paths Turn restrictions Convex polygons Convex subsets Extended formulation Dynamic programming
This work was completed while the first author was serving as Research Director of CORE, the Center for Operations Research and Econometrics, Université catholique de Louvain, whose hospitality is gratefully acknowledged. The authors would like to thank the following colleagues for useful discussions and suggestions on optimum convex polygon and convex subset problems—in approximate chronological order: Marcos Goycoolea (Universidad Adolfo Ibañez), Jeremy Barbay (Universidad de Chile), Miguel Constantino (Universidade de Lisboa), Marek Chrobak (University of California, Riverside), Keno Merckx and Jean-Paul Doignon (Université Libre de Bruxelles).
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