部分链上可能获胜者的复杂性
The Complexity of Possible Winners on Partial Chains
摘要：
可能获胜者（PW）问题是计算社会选择中的一个基本算法问题，涉及选民只表达候选人之间部分偏好的选举。通过一系列的调查，PW问题的复杂性的一个完整的分类，建立了所有纯位置评分规则：PW问题是在P的复数和否决规则，和NP完全的所有其他这样的规则。最近，PW问题被研究了在自然环境中出现的限制偏序类，如分割偏序和截断偏序;特别是，它表明，有规则的PW问题从NP完全下降到P在这样的限制偏序。在这里，我们研究部分链上的PW问题，即，偏序是其定义域的子集上的全序。这种排序在各种环境中自然出现，包括电影或餐馆的排名。我们分类的复杂性的PW问题的部分链建立，也许令人惊讶的是，这种限制并没有改变问题的复杂性，即，PW问题是NP完全的所有纯位置评分规则以外的复数和否决规则。作为一个副产品，我们得到了一个新的和更原则性的证明的复杂性的PW问题的任意偏序。
英文摘要：
The Possible Winner (PW) problem, a fundamental algorithmic problem in computational social choice, concerns elections where voters express only partial preferences between candidates. Via a sequence of investigations, a complete classification of the complexity of the PW problem was established for all pure positional scoring rules: the PW problem is in P for the plurality and veto rules, and NP-complete for all other such rules. More recently, the PW problem was studied on classes of restricted partial orders that arise in natural settings, such as partitioned partial orders and truncated partial orders; in particular, it was shown that there are rules for which the PW problem drops from NP-complete to P on such restricted partial orders. Here, we investigate the PW problem on partial chains, i.e., partial orders that are a total order on a subset of their domains. Such orders arise naturally in a variety of settings, including rankings of movies or restaurants. We classify the complexity of the PW problem on partial chains by establishing that, perhaps surprisingly, this restriction does not change the complexity of the problem, namely, the PW problem is NP-complete for all pure positional scoring rules other than the plurality and veto rules. As a byproduct, we obtain a new and more principled proof of the complexity of the PW problem on arbitrary partial orders.
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