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 only a few #P-hard problems are known to yield deterministic approximation. Motivated by the problem of deterministically approximating the volume of a $\mathcal{V}$-polytope, that is a polytope with a small number of vertices and (possibly) exponentially many facets, this paper investigates the problem of computing the volume of a “knapsack dual polytope,” which is known to be #P-hard due to Khachiyan (1989) [16]. We reduce an approximate volume of a knapsack dual polytope to that of the intersection of two cross-polytopes in a short distance, and give FPTASs for those volume computations. Interestingly, computing 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.

通常，由隶属度给定n维凸体，已知任何多项式时间确定性算法都无法近似其在比率之内的体积${（ñ/\mathrm{日志}ñ）}^{ñ}$。在诸如高维体积的马尔可夫链蒙特卡洛和许多＃P-hard问题的随机逼近方面，已经取得了实质性进展，而已知只有少数＃P-hard问题产生确定性逼近。由确定性地近似一个容器的体积的问题引起$\mathcal{V}$-polytope，即一个顶点数量少且（可能）呈指数形式的多面体，本文研究计算“背包双重多面体”的体积的问题，由于Khachiyan，它被称为＃P-hard （1989）[16]。我们在短距离内将背包双多边形的近似体积减少到两个交叉多边形的相交的体积，并为这些体积计算提供FPTAS。有趣的是，计算两个交叉多边形的相交量（即${\mathrm{大号}}_{\mathrm{1个}}$-balls）是＃P-hard，与 ${\mathrm{大号}}_{\infty }$-球或 ${\mathrm{大号}}_2$-球。