Solving Linear Programming with Constraints Unknown
What is the value of input information in solving linear programming? The celebrated ellipsoid algorithm tells us that the full information of input constraints is not necessary; the algorithm works as long as there exists an oracle that, on a proposed candidate solution, returns a violation in the form of a separating hyperplane. Can linear programming still be efficiently solved if the returned violation is in other formats?
Motivated by some real-world scenarios, we study this question in a trial-and-error framework: there is an oracle that, upon a proposed solution, returns the index of a violated constraint (with the content of the constraint still hidden). When more than one constraint is violated, two variants in the model are investigated. (1) The oracle returns the index of a “most violated” constraint, measured by the Euclidean distance of the proposed solution and the half-spaces defined by the constraints. In this case, the LP can be efficiently solved (under a mild condition of non-degeneracy). (2) The oracle returns the index of an arbitrary (i.e., worst-case) violated constraint. In this case, we give an algorithm with running time exponential in the number of variables. We then show that the exponential dependence on n is unfortunately necessary even for the query complexity. These results put together shed light on the amount of information that one needs in order to solve a linear program efficiently.
The proofs of the results employ a variety of geometric techniques, including the weighted spherical Voronoi diagram and the furthest Voronoi diagram.
- 2.Bei, X., Chen, N., Zhang, S.: On the complexity of trial and error. In: Proceedings of the 45th ACM Symposium on Theory of Computing, pp. 31–40 (2013)Google Scholar
- 3.Bei, X., Chen, N., Zhang, S.: Solving linear programming with constraints unknown (2013). arXiv:1304.1247
- 6.Doherty, L., Pister, K., Ghaoui, L.E.: Convex position estimation in wireless sensor networks. In: Proceedings of the Twentieth IEEE Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM), pp. 1655–1663 (2001)Google Scholar
- 7.Dyer, M., Megiddo, N., Welzl, E.: Linear programming. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry. CRC Press (2004)Google Scholar
- 10.Gentile, C.: Distributed sensor location through linear programming with triangle inequality constraints. In: Proceedings of IEEE Conference on Communications, pp. 3192–3196 (2005)Google Scholar
- 11.Grötschel, M., Lovász, L., Schrijver, A.: Geometric methods in combinatorial optimization. In: Progress in Combinatorial Optimization, pp. 167–183 (1984)Google Scholar
- 12.Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer (1988)Google Scholar
- 14.Ivanyos, G., Kulkarni, R., Qiao, Y., Santha, M., Sundaram, A.: On the complexity of trial and error for constraint satisfaction problems. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8572, pp. 663–675. Springer, Heidelberg (2014) Google Scholar
- 15.Kalai, G.: A subexponential randomized simplem algorithm. In: Proceedings of the ACM Symposium on Theory of Computing (STOC), pp. 475–482 (1992)Google Scholar
- 17.Kovalev, M.: A property of convex sets and its application. Matematicheskie Zametki, pp. 89–99. English translation: Mathematical Notes, 44, 537–543 (1988)Google Scholar
- 20.Montgomery, D.: Design and Analysis of Experiments, 7 edn. Wiley (2008)Google Scholar
- 21.Papadimitriou, C.H., Steiglitz, K.: Combinatorial Optimization: Algorithms and Complexity. Dover Publications (1998)Google Scholar
- 22.Papadimitriou, C.H., Yannakakis, M.: Linear programming without the matrix. In: Proceedings of the ACM Symposium on Theory of Computing (STOC), pp. 121–129 (1993)Google Scholar
- 25.Yudin, D.B., Nemirovskii, A.S.: Informational complexity and efficient methods for the solution of convex extremal problems. Ekonomika i Matematicheskie Metody, 12, 357–369 (1976). English translation: Matekon 13(3), 25–45 (1977)Google Scholar