关于全比例表示的计算
On the Computation of Fully Proportional Representation
摘要：
我们研究了Chamberlin Courant和Monroe提出的两种完全比例代表制。这两种制度都为每个选民指派一名代表，以使"虚假陈述的总和"最小化。这两个系统的获胜者确定问题是已知的NP难，因此这项工作的目的是调查是否有变种的建议规则和/或特定的选民，这些问题可以有效地解决。作为这些规则的一种变化，而不是最小化的总和误报，我们认为最大误报有效地引入两个新的规则。在一般情况下，这些经典规则的"极大极小"版本似乎仍然是NP难的。我们研究了两个经典规则和两个新规则关于多个参数的获胜者确定的参数复杂性。在这里，我们有一个积极和消极的结果的混合：例如，我们证明了固定参数的易处理性的参数的候选人的数量，但固定参数的棘手的赢家的数量。对于单峰选民，我们的结果是压倒性的积极：我们提供了多项式时间算法的大多数考虑的问题。唯一一个对单峰选民仍然是NP难的规则是经典的门罗规则。
英文摘要：
We investigate two systems of fully proportional representation suggested by Chamberlin Courant and Monroe. Both systems assign a representative to each voter so that the "sum of misrepresentations" is minimized. The winner determination problem for both systems is known to be NP-hard, hence this work aims at investigating whether there are variants of the proposed rules and/or specific electorates for which these problems can be solved efficiently. As a variation of these rules, instead of minimizing the sum of misrepresentations, we considered minimizing the maximal misrepresentation introducing effectively two new rules. In the general case these "minimax" versions of classical rules appeared to be still NP-hard. We investigated the parameterized complexity of winner determination of the two classical and two new rules with respect to several parameters. Here we have a mixture of positive and negative results: e.g., we proved fixed-parameter tractability for the parameter the number of candidates but fixed-parameter intractability for the number of winners. For single-peaked electorates our results are overwhelmingly positive: we provide polynomial-time algorithms for most of the considered problems. The only rule that remains NP-hard for single-peaked electorates is the classical Monroe rule.
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