Extended formulations for perfect domination problems and their algorithmic implications
针对完美顶点支配和完美边支配问题，提出了基于完美支配集结构特性的新数学形式，实验表明新形式在标准混合整数规划求解中能大幅提升计算速度。
Given an undirected graph G = ( V , E ) , a subset D ⊆ V is called a vertex dominating set (VDS) if every vertex of V either belongs to D or is adjacent to a vertex of D . Additionally, a VDS D is called perfect if every vertex of V ∖ D is adjacent to a single vertex of D . Finally, the Perfect Vertex Domination Problem (PVDP) asks for a perfect VDS D with the smallest cardinality possible. Domination extends very naturally to the edges of G = ( V , E ) and the Perfect Edge Domination Problem (PEDP) asks for a perfect edge dominating set with as few edges as possible. We propose new formulations for PVDP and PEDP. They rely on structural properties of perfect dominating sets and are computationally compared with their counterparts from the literature. For the new PEDP formulation, in particular, running times for standard state-of-the-art mixed integer programming codes are shown to frequently lead to speed-ups of orders of magnitude, over their corresponding performances for the remaining PEDP formulations.