The bribery problem in election has received considerable attention in the literature, upon which various algorithmic and complexity results have been obtained. It is thus natural to ask whether we can protect an election from potential bribery. We assume that the protector can protect a voter with some cost (e.g., by isolating the voter from potential bribers). A protected voter cannot be bribed. Under this setting, we consider the following bi-level decision problem: Is it possible for the protector to protect a proper subset of voters such that no briber with a fixed budget on bribery can alter the election result? The goal of this paper is to give a full picture on the complexity of protection problems. We give an extensive study on the protection problem and provide algorithmic and complexity results. Comparing our results with that on the bribery problems, we observe that the protection problem is in general significantly harder. Indeed, it becomes $\sum_{p}^2$-complete even for very restricted special cases, while most bribery problems lie in NP. However, it is not necessarily the case that the protection problem is always harder. Some of the protection problems can still be solved in polynomial time, while some of them remain as hard as the bribery problem under the same setting.

中文翻译：

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保护选举免受贿赂的计算复杂性特征

选举中的贿赂问题在文献中受到了相当大的关注，已经获得了各种算法和复杂性的结果。因此很自然地会问我们是否可以保护选举免受潜在的贿赂。我们假设保护者可以以一定的成本保护选民（例如，通过将选民与潜在的贿赂者隔离开来）。受保护的选民不能被贿赂。在这种情况下，我们考虑以下双层决策问题：保护者是否有可能保护适当的选民子集，使得具有固定贿赂预算的贿赂者无法改变选举结果？本文的目标是全面了解保护问题的复杂性。我们对保护问题进行了广泛的研究，并提供了算法和复杂性结果。将我们的结果与贿赂问题的结果进行比较，我们观察到保护问题通常要困难得多。事实上，即使对于非常有限的特殊情况，它也变成了 $\sum_{p}^2$-complete，而大多数贿赂问题在于 NP。然而，保护问题并不一定总是更难。一些保护问题仍然可以在多项式时间内解决，而一些保护问题仍然与相同设置下的贿赂问题一样困难。