We present a deterministic polynomial-time algorithm for estimating the volume of a hypercube intersected by a fixed number of constraints of the type $f(x) \leq b$, where $f$ is the sum of univariate functions that are each nonnegative, monotone, and convex. Such constraints include knapsack and norm-ball constraints. The case of the unit hypercube truncated by a single linear constraint (halfspace) is already #P-hard. Given $k$ such constraints in dimension $n$, with total input length of at most $L$ bits, total output length of at most $L_o$ bits, and an error parameter $\varepsilon > 0$, our algorithm computes a $(1 + \varepsilon)$-multiplicative approximation of the volume of their intersection with the unit hypercube $[0,1]^n$ in time poly$_k(n, 1/\varepsilon, L,L_o)$.

翻译：我们提出了一个确定性多项式时间算法，用于估计超立方体与固定数量形如$f(x) \leq b$的约束相交区域的体积，其中$f$是若干非负、单调且凸的单变量函数之和。此类约束包括背包约束和范数球约束。对于仅受单一线性约束（半空间）截断的单位超立方体，其体积计算问题已是#P-难的。给定$n$维空间中的$k$个此类约束，输入总比特数至多为$L$，输出总比特数至多为$L_o$，以及误差参数$\varepsilon > 0$，我们的算法可在时间poly$_k(n, 1/\varepsilon, L,L_o)$内，计算这些约束与单位超立方体$[0,1]^n$交集体积的$(1 + \varepsilon)$-乘性近似。