We study the integrality gap of the natural linear programming relaxation for the Bounded Color Matching (BCM) problem. We provide several families of instances and establish lower bounds on their integrality gaps and we study how the Sherali–Adams “lift-and-project” technique behaves on these instances. We complement these results by showing that if we exclude certain simple sub-structures from our input graphs, then the integrality gap of the natural linear formulation strictly improves. To prove this, we adapt for our purposes the results of Füredi (1981). We further leverage this to show upper bounds on the performance of the Sherali–Adams hierarchy when applied to the natural LP relaxation of the BCM problem.

我们研究了有界色彩匹配（BCM）问题的自然线性规划松弛的完整性缺口。我们提供了多个实例族，并在它们的完整性差距上建立了下界，我们研究了Sherali-Adams的“提升并投影”技术在这些实例上的表现。通过显示出如果我们从输入图中排除某些简单的子结构，则对这些结果进行补充，则自然线性公式的完整性差距将得到严格改善。为了证明这一点，我们将Füredi（1981）的结果用于我们的目的。当进一步应用到BCM问题的自然LP松弛时，我们进一步利用它来显示Sherali-Adams层次结构的性能上限。