By design, the Bitcoin protocol has a low throughput. The Lightning Network (LN) is a layer-two solution built to increase throughput by cryptographically securing commitments to transactions and only occasionally converting cumulative balances into on-chain transactions. LN channels enable payments between nodes connected by a path of channels. The payment flow through a channel determines its cost. Different channel topologies can support the same underlying flows but impose different costs. This paper obtains necessary conditions for cost-minimizing topologies by identifying local cost-reducing strategies. The first local strategy entails repositioning of channels. The second entails adding hubs to handle the flows of groups of nodes. The paper also evaluates the efficiency of a global configuration, obtaining bounds on the minimum cost topology and showing the unusual circumstances in which the cost minimal structure is a hub that connects to all other nodes.