用符号摄动法作简单证明
Simpler Proofs by Symbolic Perturbation
摘要：
在算法分析中，大量的精力往往花在讨论特殊情况上。例如，当分析分别考虑X和Y的情况时<Y and X>，我们可能必须特别注意当X=Y时会发生什么。另一方面，经验告诉我们，当一个尚未被重视的这类特殊情况被发现时，人们几乎总是能找到一种方法来处理它，通常是通过对分析和/或算法进行非常轻微的修改来完成的。在这篇文章中，我们从理论上证实了这一观察。我们专注于加权组合优化问题的确定性算法。这类问题实例由其结构和权重向量定义。空情况的概念被引入作为问题实例的集合，其权重向量在所有可能的权重配置的空间中构成无处开集（或空集）。一个算法被称为鲁棒的，如果任何空的情况下，可以忽略它的解决方案的质量和资源需求的分析。我们表明，实现鲁棒性只是以正确的方式打破关系的问题。更具体地说，我们表明，从几何算法领域已知的符号扰动的概念，保证不会发生任何意外的情况下，空。我们认为，对于一个巨大的类的组合优化算法，很容易验证，他们隐含地使用符号扰动打破关系，从而可以分析的假设下，一些任意空的情况下从来没有发生。最后，我们证明了存在一个符号扰动平局打破任何算法的政策。
英文摘要：
In analyses of algorithms, a substantial amount of effort has often to be spent on the discussion of special cases. For example, when the analysis considers the cases X<Y and X>Y separately, one might have to be especially careful about what happens when X=Y. On the other hand, experience tells us that when a yet unregarded special case of this kind is discovered, one nearly always finds a way to handle it. This is typically done by modifying the analysis and/or the algorithm very slightly. In this article we substantiate this observation theoretically. We concentrate on deterministic algorithms for weighted combinatorial optimization problems. A problem instance of this kind is defined by its structure and a vector of weights. The concept of a null case is introduced as set of problem instances whose weight vectors constitute a nowhere open set (or null set) in the space of all possible weight configurations. An algorithm is called robust if any null case can be disregarded in the analysis of both its solution quality and resource requirements. We show that achieving robustness is only a matter of breaking ties the right way. More specifically, we show that the concept of symbolic perturbation known from the area of geometric algorithms guarantees that no surprises will happen in null cases. We argue that for a huge class of combinatorial optimization algorithms it is easy to verify that they implicitly use symbolic perturbation for breaking ties and thus can be analyzed under the assumption that some arbitrary null case never occurs. Finally, we prove that there exists a symbolic perturbation tie breaking policy for any algorithm.
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