The “exact subgraph” approach was recently introduced as a hierarchical scheme to get increasingly tight semidefinite programming relaxations of several NP-hard graph optimization problems. Solving these relaxations is a computational challenge because of the potentially large number of violated subgraph constraints. We introduce a computational framework for these relaxations designed to cope with these difficulties. We suggest a partial Lagrangian dual, and exploit the fact that its evaluation decomposes into several independent subproblems. This opens the way to use the bundle method from non-smooth optimization to minimize the dual function. Finally computational experiments on the Max-Cut, stable set and coloring problem show the excellent quality of the bounds obtained with this approach.

中文翻译：

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基于精确子图的最大切割、稳定集和着色的 SDP 边界的计算研究

“精确子图”方法最近作为一种分层方案被引入，以获得几个 NP 难图优化问题的日益严格的半定规划松弛。由于潜在的大量违反子图约束，解决这些松弛是一个计算挑战。我们为这些松弛引入了一个计算框架，旨在应对这些困难。我们建议使用部分拉格朗日对偶，并利用其评估分解为几个独立子问题的事实。这开辟了使用非平滑优化中的捆绑方法来最小化对偶函数的方法。最后，关于 Max-Cut、稳定集和着色问题的计算实验表明，通过这种方法获得的边界具有出色的质量。