Fine Grained Lower Bounds for Multidimensional Knapsack

TL;DR

This paper establishes tight lower bounds on the computational complexity of approximation schemes and exact algorithms for the multidimensional knapsack problem, showing that current algorithms are essentially optimal under certain complexity assumptions.

Contribution

It provides the first strong lower bounds for the running times of PTAS and exact algorithms for the d-dimensional knapsack problem, matching existing upper bounds up to polylogarithmic factors.

Findings

No faster PTAS than n^{o(d/ε)} assuming Gap-ETH.

No exact algorithms faster than (n+W)^{o(d/ log d)} assuming ETH.

Approximation hardness results for constant ε and large d.

Abstract

We study the $d$-dimensional knapsack problem. We are given a set of items, each with a $d$-dimensional cost vector and a profit, along with a $d$-dimensional budget vector. The goal is to select a set of items that do not exceed the budget in all dimensions and maximize the total profit. A PTAS with running time $n^{\Theta(d/\varepsilon)}$ has long been known for this problem, where $\varepsilon$ is the error parameter and $n$ is the encoding size. Despite decades of active research, the best running time of a PTAS has remained $O(n^{\lceil d/\varepsilon \rceil - d})$. Unfortunately, existing lower bounds only cover the special case with two dimensions $d = 2$, and do not answer whether there is a $n^{o(d/\varepsilon)}$-time PTAS for larger values of $d$. The status of exact algorithms is similar: there is a simple $O(n \cdot W^d)$-time (exact) dynamic programming algorithm, where $W$ is the maximum budget, but there is no lower bound which explains the strong exponential dependence on $d$. In this work, we show that the running times of the best-known PTAS and exact algorithm cannot be improved up to a polylogarithmic factor assuming Gap-ETH. Our techniques are based on a robust reduction from 2-CSP, which embeds 2-CSP constraints into a desired number of dimensions, exhibiting tight trade-off between $d$ and $\varepsilon$ for most regimes of the parameters. Informally, we obtain the following main results for $d$-dimensional knapsack. No $n^{o(d/\varepsilon \cdot 1/(\log(d/\varepsilon))^2)}$-time $(1-\varepsilon)$-approximation for every $\varepsilon = O(1/\log d)$. No $(n+W)^{o(d/\log d)}$-time exact algorithm (assuming ETH). No $n^{o(\sqrt{d})}$-time $(1-\varepsilon)$-approximation for constant $\varepsilon$. $(d \cdot \log W)^{O(d^2)} + n^{O(1)}$-time $\Omega(1/\sqrt{d})$-approximation and a matching $n^{O(1)}$-time lower~bound.