# A Polynomial Time Algorithm for Fair Resource Allocation in Resource Exchange

## TL;DR

This paper presents a polynomial time algorithm for computing bottleneck decompositions in networks, enabling fair resource allocation in resource exchange systems with proven efficiency and fairness properties.

## Contribution

It introduces a polynomial time algorithm for bottleneck decomposition, facilitating fair resource allocation and completing the computation of market equilibrium in resource exchange.

## Key findings

- Bottleneck decomposition can be computed in O(n^6 log(nU)) time.
- Fair resource allocation can be achieved efficiently using the proposed method.
- The work completes the understanding of market equilibrium and fairness in resource exchange.

## Abstract

The rapid growth of wireless and mobile Internet has led to wide applications of exchanging resources over network, in which how to fairly allocate resources has become a critical challenge. To motivate sharing, a BD Mechanism is proposed for resource allocation, which is based on a combinatorial structure called bottleneck decomposition. The mechanism has been shown with properties of fairness, economic efficiency [17], and truthfulness against two kinds of strategic behaviors [2,3]. Unfortunately, the crux on how to compute a bottleneck decomposition of any graph is remain untouched. In this paper, we focus on the computation of bottleneck decomposition to fill the blanks and prove that the bottleneck decomposition of a network $G = (V, E; w_v)$ can be computed in $O(n^6 log(nU))$, where $n = |V|$ and $U = max_{v\in V} w_v$. Based on the bottleneck decomposition, a fair allocation in resource exchange system can be obtained in polynomial time. In addition, our work completes the computation of a market equilibrium and its relationship to two concepts of fairness in resource exchange.