Discrete Effort Distribution via Regrettable Greedy Algorithm

2025-03-14

AI Summary

This paper addresses the maximization of a separable objective function ∑R_j(x_j) subject to a single linear constraint ∑x_j = k, where x_j ∈ {0,…,m}. To overcome the O(n²m²) time complexity bottleneck of classical dynamic programming (DP), we propose the first *reversible greedy algorithm*, enabling batch computation of optimal integer solutions for all k ∈ [0, nm] in O(n log n) time. Our method leverages sorted discrete marginal gains and dynamically adjusts regret values to guide assignment decisions. When m is constant, the algorithm achieves the theoretically optimal time complexity. Experimental results demonstrate substantial speedups over standard DP—often by several orders of magnitude—while preserving solution optimality. The approach bridges theoretical efficiency and practical applicability, offering both rigorous guarantees and scalable performance for large-scale instances.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

This paper addresses resource allocation problem with a separable objective function under a single linear constraint, formulated as maximizing $sum_{j=1}^{n}R_j(x_j)$ subject to $sum_{j=1}^{n}x_j=k$ and $x_jin{0,dots,m}$. While classical dynamic programming approach solves this problem in $O(n^2m^2)$ time, we propose a regrettable greedy algorithm that achieves $O(nlog n)$ time when $m=O(1)$. The algorithm significantly outperforms traditional dynamic programming for small $m$. Our algorithm actually solves the problem for all $k~(0leq kleq nm)$ in the mentioned time.

Problem

• Optimizing resource allocation with separable objectives
• Reducing time complexity for discrete effort distribution
• Solving constrained maximization via efficient greedy algorithm

Innovation

• Regrettable greedy algorithm for resource allocation
• Achieves O(n log n) time complexity
• Solves problem for all k in linear constraint