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.

中文翻译：

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论流动民主的参数化复杂性

在流动民主中，每个选民要么自己投票，要么将自己的投票委托给其他选民。这产生了所谓的委托图。为了确定最终投票的选民及其投票的子集，我们需要解决委托图中的循环。这就产生了“解决委派”问题，在该问题中，我们需要找到委派图的非循环子图，以便其投票数的选民数量以某个整数{\ lambda}为界。对投票者所能投票的选民人数设置上限，可使系统设计人员限制任何单个投票者的权力。解决委派问题已经众所周知是NP难题。在本文中，我们研究了此问题的参数化复杂度。我们证明，对于参数{\ lambda}，宿节点数和委托图的最大程度，解析委托对para-NP困难。我们还显示，即使相对于委托图的树宽而言，“解析委托”也是W [1] -hard。我们通过展示一些其他参数的FPT算法来补充我们的负面结果。最后，我们证明了一个相关的问题，可以解决多项式时间问题，我们称之为解析分数委托。