  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 $\varepsilon >0$.
