Optimum turn-restricted paths, nested compatibility, and optimum convex polygons

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.

[1]  Gordon T. Wilfong.  Motion planning for an autonomous vehicle , 1988, Proceedings. 1988 IEEE International Conference on Robotics and Automation.

[2]  Douglas M. King,et al.  The geo-graph in practice: creating United States Congressional Districts from census blocks , 2018, Comput. Optim. Appl..

[3]  Jean-Paul Doignon,et al.  On realizable biorders and the biorder dimension of a relation , 1984 .

[4]  Gérard Cornuéjols,et al.  Extended formulations in combinatorial optimization , 2013, Ann. Oper. Res..

[5]  Jorge Urrutia,et al.  Computing optimal islands , 2011, Oper. Res. Lett..

[6]  Tom Caldwell,et al.  On finding minimum routes in a network with turn penalties , 1961, CACM.

[7]  Peter Gritzmann,et al.  Geometric Clustering for the Consolidation of Farmland and Woodland , 2014 .

[8]  Dirk Cattrysse,et al.  Compactness in Spatial Decision Support: A Literature Review , 2010, ICCSA.

[9]  László Lovász,et al.  Polyhedral Results for Antimatroids a , 1989 .

[10]  David A. Bader,et al.  Alternating criteria search: a parallel large neighborhood search algorithm for mixed integer programs , 2018, Comput. Optim. Appl..

[11]  Egon Balas,et al.  programming: Properties of the convex hull of feasible points * , 1998 .

[12]  Irene Pellegrin Llorente,et al.  Recognizing Spatial Considerations in Forest Management Planning , 2017, Current Forestry Reports.

[13]  Andrew V. Goldberg,et al.  Route Planning in Transportation Networks , 2015, Algorithm Engineering.

[14]  Fernando Ordóñez,et al.  Shape and Balance in Police Districting , 2015 .

[15]  Andrew V. Goldberg,et al.  Customizable Route Planning , 2011, SEA.

[16]  Robert Geisberger,et al.  Efficient Routing in Road Networks with Turn Costs , 2011, SEA.

[17]  Gérard Cornuéjols,et al.  Extended formulations in combinatorial optimization , 2010, 4OR.

[18]  David Eppstein,et al.  Finding minimum areak-gons , 1992, Discret. Comput. Geom..

[19]  Andrés L. Medaglia,et al.  Labeling algorithm for the shortest path problem with turn prohibitions with application to large-scale road networks , 2008, Ann. Oper. Res..