稀疏布尔二次松弛的体积计算
Volume computation for sparse boolean quadric relaxations
摘要：
受理解难处理多面体的易处理凸松弛的性质的启发，Ko等人给出了完全图$G=K_n$的布尔二次多面体（也称为（全）相关多面体）$\mathscr{P}(G)$的标准松弛$\mathscr{Q}(G)$的体积的封闭形式表达式。我们扩展这项工作的结构稀疏图，给出：（一）一个有效的算法$vol(\mathscr{Q}(G))$时$G$有界树宽度，（二）封闭形式的表达式（和渐近行为）为$vol(\mathscr{Q}(G))$的所有恒星，路径和周期，和（iii）封闭形式的表达式为$vol(\mathscr{P}(G))$的所有周期。此外，我们证明，当$G$是一个周期，简单的松弛$\mathscr{Q}(G)$是一个非常接近的模型，更复杂的$\mathscr{P}(G)$。此外，我们给出了一些计算结果表明，这种行为的周期似乎可以扩展到更复杂的图形。最后，我们推测的可能性，我们的一些结果cactii，甚至串并联图。
英文摘要：
Motivated by understanding the quality of tractable convex relaxations of intractable polytopes, Ko et al. gave a closed-form expression for the volume of a standard relaxation $\mathscr{Q}(G)$ of the boolean quadric polytope (also known as the (full) correlation polytope) $\mathscr{P}(G)$ of the complete graph $G=K_n$. We extend this work to structured sparse graphs, giving: (i) an efficient algorithm for $vol(\mathscr{Q}(G))$ when $G$ has bounded tree width, (ii) closed-form expressions (and asymptotic behaviors) for $vol(\mathscr{Q}(G))$ for all stars, paths, and cycles, and (iii) a closed-form expression for $vol(\mathscr{P}(G))$ for all cycles. Further, we demonstrate that when $G$ is a cycle, the simple relaxation $\mathscr{Q}(G)$ is a very close model for the much more complicated $\mathscr{P}(G)$. Additionally, we give some computational results demonstrating that this behavior of the cycle seems to extend to more complicated graphs. Finally, we speculate on the possibility of extending some of our results to cactii or even series-parallel graphs.
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