Fine Grained Lower Bounds for Multidimensional Knapsack

TL;DR

This work resolves the question of whether the classic PTAS for the $d$-dimensional knapsack can be substantially improved by showing strong lower bounds under Gap-ETH and ETH that tightly relate the dimension $d$ and the approximation parameter. The authors advance two central reductions from 2-CSP with rectangular constraints (R-CSP) to d-dimensional knapsack and to VK with varying dimensions, yielding near-tight trade-offs that rule out subpolynomial-time PTAS, polylogarithmic-exponent improvements, and strong exponential dependence on $d$ for exact and constant-factor approximations. They also provide a constructive $oxed{1/ ext{√d}}$-approximation algorithm that is practical under controlled budgets, complementing the hardness results and offering a clearer landscape of what is achievable in polynomial time versus under exponential-time hypotheses. Overall, the paper deepens our understanding of the complexity of high-dimensional knapsack and its relation to CSP techniques, with implications for both exact and approximate algorithm design in high dimensions.

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 $n^{Θ(d/\varepsilon)}$ 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 $Ω(1/\sqrt{d})$-approximation and a matching $n^{O(1)}$-time lower~bound.

Figures (1)

• Figure 1: An illustration of the reduction \ref{['def:DKPInstance']}. The figure shows the cost of item $i = (v,\sigma) \in I$ in dimensions $(\ell,1)$ and $(\ell,2)$ for some $\ell \in [r]$. The constraints in $D_{\ell}$ are $D_{\ell} = \{j_1,j_2,j_3,j_4\}$ where $j_1 = v$, $j_3 = e = (u,v)$ which is adjacent to $v$, and $j_2,j_4$ are constraints not involving $i$. Thus, $J(\ell,v) = 2$. The constraints are ordered by $j_1,j_2,j_3, j_4$ so that ${\textnormal{ord}}_{\ell}(j_1) = 1$, ${\textnormal{ord}}_{\ell}(j_2) = 2$, etc. The cost of $i$ in dimension $(\ell,1)$ is $w_v(i)+w_{e}(i) \cdot {\mathcal{Q}}^{3}$. Considering this cost as a base-${\mathcal{Q}}$ number, the first digit is $m$ and the third digit is $\pi_{(e,u)}(\sigma)$. This is illustrated as the gray area in the figure upper rectangle, depicting the $4$-digit number in base-${\mathcal{Q}}$. The cost of $i$ in dimension $(\ell,2)$ is $2 \cdot M-c_{(\ell,1)}(i)$ (since $J(\ell,v) = 2$) depicted as the gray area in the lower rectangle. Note that the two rectangles are not in their true proportions.

Theorems & Definitions (47)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• Definition 2.4