工程精确准阈值编辑
Engineering Exact Quasi-Threshold Editing
摘要：
准阈值图是无$\{C_4, P_4\}$的图，即，它们不包含任何四个节点的圈或路作为导出子图。本文研究了$\{C_4, P_4\}$ -free编辑问题，即寻找最少的边插入或删除次数以将输入图转换为准阈值图的问题。这个问题是NP难的，但固定参数易处理（FPT）的编辑数量，通过使用分支定界算法，并承认一个简单的整数线性规划公式（ILP）。这两种方法也适用于一般的$F$ -自由编辑问题的任何有限集的图$F$。对于FPT算法，我们引入了一个快速的启发式计算高质量的下限和改进的分支策略。对于ILP，我们设计了行生成的几种变体。我们评估了这两种方法的准阈值编辑一个大的蛋白质相似性图。对于大多数情况下，我们的优化加快FPT算法的一到三个数量级。我们使用Guidance解决的ILP的运行时间只稍微快了一点。在所有优化的情况下，FPT算法比ILP稍快，即使列出所有解决方案也是如此。此外，我们表明，几乎所有的图，解决方案的先前提出的准阈值编辑启发式QTM接近最优。
英文摘要：
Quasi-threshold graphs are $\{C_4, P_4\}$-free graphs, i.e., they do not contain any cycle or path of four nodes as an induced subgraph. We study the $\{C_4, P_4\}$-free editing problem, which is the problem of finding a minimum number of edge insertions or deletions to transform an input graph into a quasi-threshold graph. This problem is NP-hard but fixed-parameter tractable (FPT) in the number of edits by using a branch-and-bound algorithm and admits a simple integer linear programming formulation (ILP). Both methods are also applicable to the general $F$-free editing problem for any finite set of graphs $F$. For the FPT algorithm, we introduce a fast heuristic for computing high-quality lower bounds and an improved branching strategy. For the ILP, we engineer several variants of row generation. We evaluate both methods for quasi-threshold editing on a large set of protein similarity graphs. For most instances, our optimizations speed up the FPT algorithm by one to three orders of magnitude. The running time of the ILP, that we solve using Gurobi, becomes only slightly faster. With all optimizations, the FPT algorithm is slightly faster than the ILP, even when listing all solutions. Additionally, we show that for almost all graphs, solutions of the previously proposed quasi-threshold editing heuristic QTM are close to optimal.
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