Because edge modification problems are computationally difficult for most target graph classes, considerable attention has been devoted to inclusion-minimal edge modifications, which are usually polynomial-time computable and which can serve as an approximation of minimum cardinality edge modifications, albeit with no guarantee on the cardinality of the resulting modification set. Over the past fifteen years, the primary design approach used for inclusion-minimal edge modification algorithms is based on a specific incremental scheme. Unfortunately, nothing guarantees that the set $\mathcal{E}$ of edge modifications of a graph G that can be obtained in this specific way spans all the inclusion-minimal edge modifications of G. Here, we focus on edge modification problems into the class of chordal graphs and we show that for this the set $\mathcal{E}$ may not even contain any solution of minimum size and may not even contain a solution close to the minimum; in fact, we show that it may not contain a solution better than within an $\mathrm{\Omega }(n)$ factor of the minimum. These results show strong limitations on the use of the current favored algorithmic approach to inclusion-minimal edge modification in heuristics for computing a minimum cardinality edge modification. They suggest that further developments might be better using other approaches.

因为对于大多数目标图类来说边修改问题在计算上是困难的，所以相当多的注意力集中在包含最小边修改上，这些边修改通常是多项式时间可计算的，可以作为最小基数边修改的近似值，尽管不能保证结果修改集的基数。在过去的十五年中，用于包含最小边缘修改算法的主要设计方法是基于特定的增量方案。不幸的是，没有什么能保证集合$\mathcal{乙}$可以通过这种特定方式获得的图G的边修改的数量涵盖G 的所有包含最小边修改。在这里，我们将边缘修改问题集中在弦图类中，并且我们证明了对于这个集合$\mathcal{乙}$甚至可能不包含任何最小尺寸的解决方案，甚至可能不包含接近最小尺寸的解决方案；事实上，我们表明它可能没有比在一个$\mathrm{\Omega }(n)$最小的因素。这些结果表明，在启发式算法中使用当前偏爱的算法方法进行包含最小边缘修改以计算最小基数边缘修改具有很大的局限性。他们建议使用其他方法进行进一步的开发可能会更好。