Incremental-Decremental Maximization

TL;DR

This paper presents a new framework for incremental-decremental maximization, focusing on transforming solutions gradually while maintaining high utility, with algorithms that perform well for submodular and related functions.

Contribution

It introduces a novel incremental-decremental maximization framework and algorithms that ensure high utility during transformations for submodular and related functions.

Findings

Algorithms maintain high utility during transformations.

Performance guarantees for submodular and gross substitute functions.

Incremental-decremental maximization is more challenging than incremental maximization.

Abstract

We introduce a framework for incremental-decremental maximization that captures the gradual transformation or renewal of infrastructures. In our model, an initial solution is transformed one element at a time and the utility of an intermediate solution is given by the sum of the utilities of the transformed and untransformed parts. We propose a simple randomized and a deterministic algorithm that both find an order in which to transform the elements while maintaining a large utility during all stages of transformation, relative to an optimum solution for the current stage. More specifically, our algorithms yield competitive solutions for utility functions of bounded curvature and/or generic submodularity ratio, and, in particular, for submodular functions, and gross substitute functions. Our results exhibit that incremental-decremental maximization is substantially more difficult than incremental maximization.