M-convex Function Minimization Under L1-Distance Constraint

Abstract

In this paper we consider a new problem of minimizing an M-convex function under L1-distance constraint (MML1); the constraint is given by an upper bound for L1-distance between a feasible solution and a given "center." This is motivated by a nonlinear integer programming problem for re-allocation of dock capacity in a bike sharing system discussed by Freund et al. (2017). The main aim of this paper is to better understand the combinatorial structure of the dock re-allocation problem through the connection with M-convexity, and show its polynomial-time solvability using this connection. For this, we first show that the dock re-allocation problem can be reformulated in the form of (MML1). We then present a pseudo-polynomial-time algorithm for (MML1) based on steepest descent approach. We also propose two polynomial-time algorithms for (MML1) by replacing the L1-distance constraint with a simple linear constraint. Finally, we apply the results for (MML1) to the dock re-allocation problem to obtain a pseudo-polynomial-time steepest descent algorithm and also polynomial-time algorithms for this problem. The proposed algorithm is based on a proximity-scaling algorithm for a relaxation of the dock re-allocation problem, which is of interest in its own right.