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M-Convex Function Minimization Under L1-Distance Constraint and Its Application to Dock Reallocation in Bike-Sharing System

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MATHEMATICS OF OPERATIONS RESEARCH 2022年第2期47卷 1566-1611页

作者： Shioura, Akiyoshi Tokyo Inst Technol Dept Ind Engn & Econ Tokyo 1528550 Japan

In this paper, we consider a problem of minimizing an M-convex function under an 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 reallocation 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 reallocation problem through the connection with M-convexity and show its polynomial-time solvability using this connection. For this, we first show that the dock reallocation problem and its generalizations can be reformulated in the form of (MML1). We then present a pseudo-polynomial-time algorithm for (MML1) based on the 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 reallocation problem to obtain a pseudo-polynomial-time steepest descent algorithm and also polynomial-time algorithms for this problem. For this purpose, we develop a polynomial-time algorithm for a relaxation of the dock reallocation problem by using a proximity-scaling approach, which is of interest in its own right.

关键词： discrete convex function discrete convex analysis resource allocation problem steepest descent algorithm proximity-scaling algorithm

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