${\cal V}$ -多胞形体积的FPTAS--两个交叉多胞形交线体积的计算困难
An FPTAS for the Volume of a ${\cal V}$-polytope ---It is Hard to Compute The Volume of The Intersection of Two Cross-polytopes
摘要：
一般情况下，给定一个$n$维凸体，已知任何多项式时间确定性算法都不能在比率$(n/\log n)^n$内近似其体积。随机近似算法如马尔可夫链蒙特卡罗算法在高维问题和许多P-难问题上取得了很大的进展，而确定性近似算法仅在少数P-难问题上得到了发展。受${\cal V}$ -多面体体积的确定性近似的启发，本文研究了“背包对偶多面体”的体积，该多面体是由Khachiyan（1989）提出的一种P-困难多面体。我们减少了近似体积的背包对偶多面体的交叉多面体的交叉，并给出FPTAS这些体积计算。有趣的是，两个交叉多面体（即，与$L_{\infty}$ -balls或$L_2$ -balls的情况不同，$L_1$ -balls）是#P-硬的。
英文摘要：
Given an $n$-dimensional convex body by a membership oracle in general, it is known that any polynomial-time deterministic algorithm cannot approximate its volume within ratio $(n/\log n)^n$. There is a substantial progress on randomized approximation such as Markov chain Monte Carlo for a high-dimensional volume, and for many #P-hard problems, while some deterministic approximation algorithms are recently developed only for a few #P-hard problems. Motivated by a deterministic approximation of the volume of a ${\cal V}$-polytope, that is a polytope with few vertices and (possibly) exponentially many facets, this paper investigates the volume of a "knapsack dual polytope," which is known to be #P-hard due to Khachiyan (1989). We reduce an approximate volume of a knapsack dual polytope to that of the intersection of two cross-polytopes, and give FPTASs for those volume computations. Interestingly, the volume of the intersection of two cross-polytopes (i.e., $L_1$-balls) is #P-hard, unlike the cases of $L_{\infty}$-balls or $L_2$-balls.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
