Weighted Packet Selection for Rechargeable Links: Complexity and Approximation

TL;DR

This paper studies a complex weighted packet selection problem for rechargeable links, proving NP-hardness and providing an efficient approximation algorithm with a specific ratio, relevant for applications like cryptocurrency networks.

Contribution

It introduces a new NP-hard problem of weighted packet selection on rechargeable links and offers an efficient approximation algorithm with a provable ratio.

Findings

The problem is NP-hard.

An approximation algorithm with ratio (1+ε)(1+√3) is developed.

The model applies to cryptocurrency network scenarios.

Abstract

We consider a natural problem dealing with weighted packet selection across a rechargeable link, which e.g., finds applications in cryptocurrency networks. The capacity of a link $(u,v)$ is determined by how much players $u$ and $v$ allocate for this link. Specifically, the input is a finite ordered sequence of packets that arrive in both directions along a link. Given $(u, v)$ and a packet of weight $x$ going from $u$ to $v$, player $u$ can either accept or reject the packet. If player $u$ accepts the packet, their capacity on link $(u,v)$ decreases by $x$. Correspondingly, player $v$ capacity on $(u,v)$ increases by $x$. If a player rejects the packet, this will entail a cost linear in the weight of the packet. A link is "rechargeable" in the sense that the total capacity of the link has to remain constant, but the allocation of capacity at the ends of the link can depend arbitrarily on players' decisions. The goal is to minimise the sum of the capacity injected into the link and the cost of rejecting packets. We show the problem is NP-hard, but can be approximated efficiently with a ratio of $(1+ \varepsilon)\cdot (1+\sqrt{3})$ for some arbitrary $\varepsilon >0$.