Computer Science > Computational Complexity
[Submitted on 30 Sep 2022 (v1), last revised 12 Sep 2024 (this version, v3)]
Title:Pure-Circuit: Tight Inapproximability for PPAD
View PDF HTML (experimental)Abstract:The current state-of-the-art methods for showing inapproximability in PPAD arise from the $\varepsilon$-Generalized-Circuit ($\varepsilon$-GCircuit) problem. Rubinstein (2018) showed that there exists a small unknown constant $\varepsilon$ for which $\varepsilon$-GCircuit is PPAD-hard, and subsequent work has shown hardness results for other problems in PPAD by using $\varepsilon$-GCircuit as an intermediate problem.
We introduce Pure-Circuit, a new intermediate problem for PPAD, which can be thought of as $\varepsilon$-GCircuit pushed to the limit as $\varepsilon \rightarrow 1$, and we show that the problem is PPAD-complete. We then prove that $\varepsilon$-GCircuit is PPAD-hard for all $\varepsilon < 0.1$ by a reduction from Pure-Circuit, and thus strengthen all prior work that has used GCircuit as an intermediate problem from the existential-constant regime to the large-constant regime.
We show that stronger inapproximability results can be derived by reducing directly from Pure-Circuit. In particular, we prove tight inapproximability results for computing approximate Nash equilibria and approximate well-supported Nash equilibria in graphical games, for finding approximate well-supported Nash equilibria in polymatrix games, and for finding approximate equilibria in threshold games.
Submission history
From: Alexandros Hollender [view email][v1] Fri, 30 Sep 2022 00:25:04 UTC (44 KB)
[v2] Fri, 3 Mar 2023 15:41:21 UTC (46 KB)
[v3] Thu, 12 Sep 2024 11:39:17 UTC (55 KB)
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