Satisfactory Budget Division

TL;DR

This work introduces a divisible-budget allocation model where each agent's demands for $m$ projects determine local and global satisfaction via a threshold $\tau$. It provides four complementary investigations: guaranteed fractions of satisfied agents, classes of instances where all agents can be satisfied, complexity of achieving universal satisfaction for a given instance, and minimizing the total budget required to satisfy all agents. The results include tight bounds for $\tau=\{1, m/2, m-1, m\}$, NP-hardness and pseudopolynomial algorithms depending on the demand structure, and a polynomial-time DP for maximizing the total number of satisfied agent-project pairs. These findings advance understanding of when broad agent satisfaction is feasible, guiding algorithmic design for participatory budgeting and related resource-allocation problems.

Abstract

A divisible budget must be allocated to several projects, and agents are asked for their opinion on how much they would give to each project. We consider that an agent is satisfied by a division of the budget if, for at least a certain predefined number $τ$ of projects, the part of the budget actually allocated to each project is at least as large as the amount the agent requested. The objective is to find a budget division that ``best satisfies'' the agents. In this context, different problems can be stated and we address the following ones. We study $(i)$ the largest proportion of agents that can be satisfied for any instance, $(ii)$ classes of instances admitting a budget division that satisfies all agents, $(iii)$ the complexity of deciding if, for a given instance, every agent can be satisfied, and finally $(iv)$ the question of finding, for a given instance, the smallest total budget to satisfy all agents. We provide answers to these complementary questions for several natural values of the parameter $τ$, capturing scenarios where we seek to satisfy for each agent all; almost all; half; or at least one of her requests.

Figures (2)

• Figure 1: Illustration of Theorem \ref{['thm3']}
• Figure 2: Remaining indices of $Q_2$, $Q_3$, $Q_1$ and $Q_4$ are represented as circles, triangles, squares, and stars, respectively. The dashed rectangles give the proposed matching

Theorems & Definitions (38)

• proof
• Definition 1
• Theorem 1
• proof
• Lemma 1
• proof
• Theorem 2
• proof
• Proposition 1
• proof