Fine Grained Lower Bounds for Multidimensional Knapsack
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arXiv
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| Format: | Preprint |
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2024
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| author | Doron-Arad, Ilan Kulik, Ariel Manurangsi, Pasin |
| author_facet | Doron-Arad, Ilan Kulik, Ariel Manurangsi, Pasin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10146 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fine Grained Lower Bounds for Multidimensional Knapsack Doron-Arad, Ilan Kulik, Ariel Manurangsi, Pasin Data Structures and Algorithms 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. |
| title | Fine Grained Lower Bounds for Multidimensional Knapsack |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2407.10146 |