Efficient solutions to relaxations of combinatorial problems with submodular penalties via the Lovász extension and non-smooth convex optimization

• Authors: Fabián A. Chudak;Kiyohito Nagano
• Venue: SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
• Year: 2007

We consider convex relaxations for combinatorial optimization problems with submodular penalties. The relaxations are obtained very naturally through a novel use of the Lovász extension of a submodular function. We also propose the use of simple and recent algorithms for non-smooth convex optimization due to Nesterov to approximately solve them. For the uncapacitated facility location problem with submodular penalties we design FPTAS for our relaxation that can be used to design approximation algorithms for the original problem for the metric case. In contrast to previous work of Hayrapetyan et al, our algorithms are fast and simple and do not use the ellipsoid algorithm. We also consider more general set covering problems with submodular costs also introduced by Hayrapetyan et al and submodular function minimization. The running time dependence on ε of our algorithms is either 1/ε or 1/ε2. The slower ones (1/ε2) are very simple. The faster ones (1/ε) are more sophisticated and require that certain projections be easy to solve in the base polytope. These projections can generally be solved using submodular function minimization as a subroutine; however for some important special cases, they can be solved much more efficiently, thus providing the fastest of our algorithms.