For approval-based liquid democracy with voting and delegation costs, finding a budget-feasible representative set is polynomial, while bounded path length, bounded voting power, and voter-control variants become NP-hard once voters may approve two or more delegates.
Alouf-Heffetz, T
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The Cost Perspective of Liquid Democracy: Feasibility and Control
For approval-based liquid democracy with voting and delegation costs, finding a budget-feasible representative set is polynomial, while bounded path length, bounded voting power, and voter-control variants become NP-hard once voters may approve two or more delegates.