Given a linear ordering of the vertices of a graph, the cutwidth of a vertex $v$ with respect to this ordering is the number of edges from any vertex before $v$ (including $v$) to any vertex after $v$ in this ordering. The cutwidth of an ordering is the maximum cutwidth of any vertex with respect to this ordering. We are interested in finding the cutwidth of a graph, that is, the minimum cutwidth over all orderings, which is an NP-hard problem. In order to approximate the cutwidth of a given graph, we present a semidefinite relaxation. We identify several classes of valid inequalities and equalities that we use to strengthen the semidefinite relaxation. These classes are on the one hand the well-known 3-dicycle equations and the triangle inequalities and on the other hand we obtain inequalities from the squared linear ordering polytope and via lifting the linear ordering polytope. The solution of the semidefinite program serves to obtain a lower bound and also to construct a feasible solution and thereby having an upper bound on the cutwidth. In order to evaluate the quality of our bounds, we perform numerical experiments on graphs of different sizes and densities. It turns out that we produce high quality bounds for graphs of medium size independent of their density in reasonable time. Compared to that, obtaining bounds for dense instances of the same quality is out of reach for solvers using integer linear programming techniques.

翻译：给定图顶点的一个线性排序，顶点$v$相对于该排序的割宽是指：在该排序中，从$v$之前（包括$v$）的任意顶点到$v$之后任意顶点的边数。一个排序的割宽是该排序中所有顶点割宽的最大值。我们关注的是求解图的割宽，即所有排序中割宽的最小值，这是一个NP难问题。为了近似计算给定图的割宽，我们提出了一种半定松弛方法。我们识别了若干类有效的约束等式与不等式，并利用它们来强化该半定松弛。这些约束类别一方面包括经典的三圈方程和三角不等式，另一方面我们从平方线性排序多面体以及通过提升线性排序多面体获得了不等式。半定规划的解可用于获得下界，同时也能构造可行解，从而得到割宽的上界。为评估所提界的质量，我们在不同规模和密度的图上进行了数值实验。结果表明，我们能在合理时间内为中等规模、任意密度的图生成高质量界。相比之下，采用整数线性规划技术的求解器无法对相同质量的稠密实例获得界。