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 $\varepsilon >0$.

翻译：我们考虑一个自然的问题,涉及在可再补给的链接中选择加权包件,例如,在加密货币网络中找到应用程序。一个链接$(u,v)美元的能力取决于为该链接分配多少玩家$(u,v)美元和美元(v)美元。具体地说,输入是一个固定的定序包件序列,该包件沿着一个链接双向抵达。鉴于美元(u,v)美元和一包重美元从1美元到5美元,玩家可以接受或拒绝该包件。如果玩家美元(u,v)美元接受该包件,那么其在链接上的能力将减少0.0美元。相应地,玩家美元(u,v)美元的能力将增加0.x美元。如果玩家拒绝这个定序包件,则包件重量将产生成本线性线性。一个链接“可负担费用”,因为连接的总能力必须保持不变,但该链接结束时的能力分配可以任意地取决于玩家的决定。这个目标是要将能力总额(xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx