Incremental-Decremental Maximization

2025-08-20

AI Summary

This work addresses the incremental-decremental optimization problem in progressive infrastructure updates: ensuring that every intermediate solution—obtained by element-wise replacement of an initial solution—maintains high utility relative to the stage-optimal solution. For utility functions exhibiting generalized bounded curvature and generic submodularity ratios—including submodular and gross substitutes functions—we propose the first systematic algorithmic frameworks, both randomized and deterministic. We rigorously characterize the competitive ratio of incremental-decremental processes and prove their computational complexity is strictly higher than that of purely incremental settings. Our algorithms provide stage-wise theoretical guarantees: achieving tight approximation ratios for both submodular and quasi-submodular functions. This establishes the first optimization paradigm for dynamic infrastructure upgrading with provable utility guarantees across the entire update trajectory.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• Develops incremental-decremental framework for infrastructure transformation optimization
• Proposes algorithms maintaining high utility during element-by-element transformation
• Addresses challenges with submodular functions and bounded curvature utilities

Innovation

• Incremental-decremental framework for infrastructure transformation
• Randomized and deterministic algorithms for element ordering
• Handles submodular functions with bounded curvature properties