Near Optimal Constant Inapproximability under ETH for Fundamental Problems in Parameterized Complexity
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3717823.3718257.pdf
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Author(s)
Bafna, Mitali
Karthik C. S.
Minzer, Dor
Date Issued
June 15, 2025
Publisher
ACM|Proceedings of the 57th Annual ACM Symposium on Theory of Computing
Citation
Mitali Bafna, Karthik C. S., and Dor Minzer. 2025. Near Optimal Constant Inapproximability under ETH for Fundamental Problems in Parameterized Complexity. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing (STOC '25). Association for Computing Machinery, New York, NY, USA, 2118–2129.
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Final published version
Abstract
We prove that under the Exponential Time Hypothesis (ETH), for every ε > 0, there exists a constant C > 0 such that no algorithm running in time nk / logC k can determine whether a given 2-CSP instance with k variables, O(k) constraints, and alphabet size n, is perfectly satisfiable or if every assignment satisfies at most an ε fraction of the constraints.
By known reductions in the literature, the above result implies near-optimal conditional lower bounds for approximating a host of parameterized problems, such as the k-Clique problem, k-Max-Coverage problem, k-Unique Set Cover problem, k-Median and k-Means problems, parameterized variants of the Nearest Codeword problem, Minimum Distance of a Code problem, Closest Vector problem, and Shortest Vector problem.
We also establish a densification theorem for the parameterized 2-CSP problem, showing that the aforementioned conditional lower bound for sparse 2-CSPs also holds when the constraint graph is a complete graph. From this densification, we conclude that assuming ETH, there is no algorithm running in time n√k / logC k that approximates the k-Directed Steiner Network problem and the k-Strongly Connected Steiner Subgraph problem to some constant factors.
Description
Mitali Bafna, Karthik C. S., and Dor Minzer. 2025. Near Optimal Constant Inapproximability under ETH for Fundamental Problems in Parameterized Complexity. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing (STOC '25). Association for Computing Machinery, New York, NY, USA, 2118–2129.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1145/3717823.3718257
