Minimizing Discrete Convex Functions with Linear Inequality Constraints

A class of discrete convex functions that can efficiently be minimized has been considered by Murota. Among them are L\-convex functions, which are natural extensions of submodular set functions. We first consider the problem of minimizing an L\-convex function with a linear inequality constraint having a positive normal vector. We propose a polynomial algorithm to solve it based on a binary search for an optimal Lagrange multiplier, where use is made of algorithms for minimum-ratio and maximum-ratio problems that are, respectively, associated with submodular and supermodular set functions. We also examine an extension of the problem to that with a linear inequality constraint having a not necessarily positive normal vector and adapt it to the problem of minimizing an M\-convex function, the convex conjugate of an L\-convex function, with a linear inequality constraint. The former extension can be solved in polynomial time by using a binary search for an optimal Lagrange multiplier and by adopting Nagano’s algorithm for the intersection of line and a base polyhedron. The latter can also be solved in polynomial time by an approach similar to that for L\-convex functions, based on a geometric characterization of M\-convex functions.