Proximity theorems of discrete convex functions

Aproximity theorem is astatement that, given an optimization problem and its relaxation, an optimal solution to the original problem exists in acertain neighborhood of asolution to the relaxation. Proximity theorems have been used successfully, for example, in designing efficient algorithms for discrete resource allocation problems. After reviewing the recent results for $\mathrm{L}$-convex and $\mathrm{M}$-convex functions, this paper establishes proximity theorems for larger classes of discrete convex functions, $\mathrm{L}_{2}$-convex functions and $\mathrm{M}_{2}$-convex functions, that are relevant to the polymatroid intersection problem and the submodular flow problem. 1Introduction In the area of discrete optimization, nonlinear optimization problems have been investigated as well as linear optimization problems. Submodular (set) functions and separable convex functions are well-known examples of tractable nonlinear functions, in that the sub-modular function minimization problem can be solved in polynomial time (see [13, 14, 24]), and separable convex functions have been treated successfully in many different discrete optimization problems (see [11]). Recently, certain classes of " discrete convex functions " were proposed: