On Parameterized Complexity of Liquid Democracy

TL;DR

This paper investigates the parameterized complexity of the Resolve Delegation problem in liquid democracy, revealing its computational hardness under various parameters and providing fixed-parameter tractable algorithms for some cases.

Contribution

The paper establishes para-NP-hardness and W[1]-hardness results for Resolve Delegation, and presents FPT algorithms for certain parameters, advancing understanding of its computational complexity.

Findings

Resolve Delegation is para-NP-hard with respect to several parameters.

Resolve Delegation is W[1]-hard with respect to treewidth.

Resolve Fractional Delegation can be solved in polynomial time.

Abstract

In liquid democracy, each voter either votes herself or delegates her vote to some other voter. This gives rise to what is called a delegation graph. To decide the voters who eventually votes along with the subset of voters whose votes they give, we need to resolve the cycles in the delegation graph. This gives rise to the Resolve Delegation problem where we need to find an acyclic sub-graph of the delegation graph such that the number of voters whose votes they give is bounded above by some integer {\lambda}. Putting a cap on the number of voters whose votes a voter gives enable the system designer restrict the power of any individual voter. The Resolve Delegation problem is already known to be NP-hard. In this paper we study the parameterized complexity of this problem. We show that Resolve Delegation is para-NP-hard with respect to parameters {\lambda}, number of sink nodes and the maximum degree of the delegation graph. We also show that Resolve Delegation is W[1]-hard even with respect to the treewidth of the delegation graph. We complement our negative results by exhibiting FPT algorithms with respect to some other parameters. We finally show that a related problem, which we call Resolve Fractional Delegation, is polynomial time solvable.