  Blockchain technology, while revolutionary in enabling decentralized
transactions, faces scalability challenges as the ledger must be replicated
across all nodes of the chain, limiting throughput and efficiency. Sharding,
which divides the chain into smaller segments, called shards, offers a solution
by enabling parallel transaction processing. However, sharding introduces new
complexities, notably how to allocate nodes to shards without compromising the
network's security.
  This paper introduces a novel linear optimization framework for node
allocation to shards that addresses decentralization constraints while
minimizing resource consumption. In contrast to traditional methods that depend
on random or trust-based assignments, our approach evaluates node
characteristics, including ownership, hardware, and geographical distribution,
and requires an explicit specification of decentralization targets with respect
to these characteristics. By employing linear optimization, the framework
identifies a resource-efficient node set meeting these targets. Adopted by the
Internet Computer Protocol (ICP) community, this framework proves its utility
in real-world blockchain applications. It provides a quantitative tool for node
onboarding and offboarding decisions, balancing decentralization and resource
considerations.
