Deterministic Volume Estimation of Truncated Hypercubes

TL;DR

This paper introduces a deterministic polynomial-time algorithm to approximate the volume of hypercube intersections with convex constraints, including knapsack and norm-ball constraints, within a specified error margin.

Contribution

It provides the first polynomial-time deterministic algorithm for volume estimation of hypercube intersections under multiple convex constraints.

Findings

Algorithm computes a (1 + ε)-approximation of volume.

Runs in polynomial time with respect to input size and inverse error.

Applicable to constraints like knapsack and norm-balls.

Abstract

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$_k(n, 1/\varepsilon, L,L_o)$.