Almost Polynomial Factor Inapproximability for Parameterized k-Clique
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866913665382350848 |
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| author | S., Karthik C. Khot, Subhash |
| author_facet | S., Karthik C. Khot, Subhash |
| contents | The k-Clique problem is a canonical hard problem in parameterized complexity. In this paper, we study the parameterized complexity of approximating the k-Clique problem where an integer k and a graph G on n vertices are given as input, and the goal is to find a clique of size at least k/F(k) whenever the graph G has a clique of size k. When such an algorithm runs in time T(k)poly(n) (i.e., FPT-time) for some computable function T, it is said to be an F(k)-FPT-approximation algorithm for the k-Clique problem.
Although, the non-existence of an F(k)-FPT-approximation algorithm for any computable sublinear function F is known under gap-ETH [Chalermsook et al., FOCS 2017], it has remained a long standing open problem to prove the same inapproximability result under the more standard and weaker assumption, W[1]$\neq$FPT.
In a recent breakthrough, Lin [STOC 2021] ruled out constant factor (i.e., F(k)=O(1)) FPT-approximation algorithms under W[1]$\neq$FPT. In this paper, we improve this inapproximability result (under the same assumption) to rule out every $F(k)=k^{1/H(k)}$ factor FPT-approximation algorithm for any increasing computable function H (for example $H(k)=\log^\ast k$).
Our main technical contribution is introducing list decoding of Hadamard codes over large prime fields into the proof framework of Lin. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_03983 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Almost Polynomial Factor Inapproximability for Parameterized k-Clique S., Karthik C. Khot, Subhash Computational Complexity The k-Clique problem is a canonical hard problem in parameterized complexity. In this paper, we study the parameterized complexity of approximating the k-Clique problem where an integer k and a graph G on n vertices are given as input, and the goal is to find a clique of size at least k/F(k) whenever the graph G has a clique of size k. When such an algorithm runs in time T(k)poly(n) (i.e., FPT-time) for some computable function T, it is said to be an F(k)-FPT-approximation algorithm for the k-Clique problem. Although, the non-existence of an F(k)-FPT-approximation algorithm for any computable sublinear function F is known under gap-ETH [Chalermsook et al., FOCS 2017], it has remained a long standing open problem to prove the same inapproximability result under the more standard and weaker assumption, W[1]$\neq$FPT. In a recent breakthrough, Lin [STOC 2021] ruled out constant factor (i.e., F(k)=O(1)) FPT-approximation algorithms under W[1]$\neq$FPT. In this paper, we improve this inapproximability result (under the same assumption) to rule out every $F(k)=k^{1/H(k)}$ factor FPT-approximation algorithm for any increasing computable function H (for example $H(k)=\log^\ast k$). Our main technical contribution is introducing list decoding of Hadamard codes over large prime fields into the proof framework of Lin. |
| title | Almost Polynomial Factor Inapproximability for Parameterized k-Clique |
| topic | Computational Complexity |
| url | https://arxiv.org/abs/2112.03983 |