发表机构
University of Michigan（密歇根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对级联图，提出基于级联分解的动态规划算法，精确计算布尔二次多胞形体积，复杂度为 O(d^7)，并分析其与线性松弛的体积比率及奇环不等式的影响。
AI 中文摘要
对于图 G，布尔二次多胞形 P(G) 是满足 $y_{ij}=x_ix_j$（其中 $ij\in E(G)$）的二元解的凸包，而 Q(G) 是其标准线性松弛。对于级联图，Q(G) 与奇环不等式一起描述了 P(G)。Lee 和 Skipper 证明了对于有界树宽，vol(Q(G)) 可在多项式时间内计算，并给出了当 G 为环时 vol(P(G)) 的闭式公式。我们解决了他们提出的关于在级联图上给出 vol(P(G)) 的高效算法的问题。设 $d=|V(G)|+|E(G)|$，我们用 $O(d^7)$ 次算术运算计算它，若 G 是仙人掌图则用 $O(d^5)$ 次。该算法是基于级联分解的动态规划。同一框架用 $O(d^4)$ 次运算计算 vol(Q(G))（因此也计算 G 的关联偏序集的线性扩展数），对仙人掌图则为 $O(d^3)$ 次。当每个 $x_v$ 固定为 1/2 时，递归简化为单变量多项式的卷积，并以 $O(m^3)$ 次运算计算每个具有 m 条边的级联图的割多胞形体积。我们还研究了 P(G) 占据 Q(G) 的多少比例。短奇环比长奇环更重要：在每个图中，长度为 $\ell$ 的环的奇环不等式最多切掉 Q(G) 的 $2^{\ell-1}/\ell!$ 部分。因此，对于级联图 G，vol(P(G))/vol(Q(G)) $\ge 1-\sum_C 2^{|C|-1}/|C|!$，其中 C 遍历其所有环。该比率不因共享顶点的环而分解：在由 $C_\ell$ 的 k 个副本组成的花图中，它按 $\rho_\ell^k$ 衰减，其中 $\rho_\ell$ 是低于 $C_\ell$ 比率的显式有理数。然而，许多长环切掉的小部分会累积：在最坏情况下，三角形不等式或任意固定长度的奇环不等式不能闭合 Q(G) 与 P(G) 之间间隙的固定部分，且体积比率可能随 d 呈指数级小。
英文摘要
For a graph G, the Boolean quadric polytope P(G) is the convex hull of the binary solutions of $y_{ij}=x_ix_j$ for $ij\in E(G)$, and Q(G) is its standard linear relaxation. For series-parallel graphs, Q(G) together with the odd-cycle inequalities describes P(G). Lee and Skipper showed vol(Q(G)) is polynomial-time computable for bounded treewidth and gave a closed formula for vol(P(G)) when G is a cycle. We resolve their question of giving an efficient algorithm for vol(P(G)) on series-parallel graphs. With $d=|V(G)|+|E(G)|$, we compute it with $O(d^7)$ arithmetic operations, and $O(d^5)$ if G is a cactus. The algorithm is a dynamic program over the series-parallel decomposition. The same framework computes vol(Q(G)) (hence the number of linear extensions of the incidence poset of G) with $O(d^4)$ operations, and $O(d^3)$ for cacti. With every $x_v$ fixed at 1/2, the recursion reduces to convolutions of univariate polynomials and computes the cut polytope volume of every series-parallel graph with m edges in $O(m^3)$ operations. We also study how much of Q(G) the polytope P(G) occupies. Short odd cycles matter more than long ones: in every graph, the odd-cycle inequalities of a cycle of length $\ell$ cut off at most a fraction $2^{\ell-1}/\ell!$ of Q(G). Hence vol(P(G))/vol(Q(G)) $\ge 1-\sum_C 2^{|C|-1}/|C|!$ for series-parallel G, with C ranging over its cycles. The ratio does not factor over cycles sharing a vertex: in a flower of $k$ copies of $C_\ell$ it decays like $ρ_\ell^k$ for an explicit rational $ρ_\ell$ below the ratio of $C_\ell$. Nevertheless, the small fractions cut off by many long cycles compound: in the worst case, triangle inequalities, or odd-cycle inequalities up to any fixed length, close no fixed fraction of the gap between Q(G) and P(G), and the volume ratio can be exponentially small in d.
