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Pure-Circuit: Tight Inapproximability for PPAD

Argyrios Deligkas, John Fearnley, Alexandros Hollender, Themistoklis Melissourgos

TL;DR

The paper introduces Pure-Circuit, a two-gate PPAD-complete intermediate problem that captures the essential hardness structure needed for large-constant inapproximability results. By reducing from Pure-Circuit, the authors derive tight, constant-factor hardness results for computing approximate Nash equilibria and well-supported equilibria across polymatrix, graphical, and threshold games, improving prior existential-constant bounds. The core approach leverages a PURIFY gate and robustness to simulate strong nontrivial constraints, enabling a direct and modular reduction path to multiple PPAD problems. This work thus provides a clean, unified framework to obtain large-constant PPAD hardness results and identifies Pure-Circuit as the canonical intermediate problem for future PPAD inapproximability research.

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.

Pure-Circuit: Tight Inapproximability for PPAD

TL;DR

The paper introduces Pure-Circuit, a two-gate PPAD-complete intermediate problem that captures the essential hardness structure needed for large-constant inapproximability results. By reducing from Pure-Circuit, the authors derive tight, constant-factor hardness results for computing approximate Nash equilibria and well-supported equilibria across polymatrix, graphical, and threshold games, improving prior existential-constant bounds. The core approach leverages a PURIFY gate and robustness to simulate strong nontrivial constraints, enabling a direct and modular reduction path to multiple PPAD problems. This work thus provides a clean, unified framework to obtain large-constant PPAD hardness results and identifies Pure-Circuit as the canonical intermediate problem for future PPAD inapproximability research.

Abstract

The current state-of-the-art methods for showing inapproximability in PPAD arise from the -Generalized-Circuit (-GCircuit) problem. Rubinstein (2018) showed that there exists a small unknown constant for which -GCircuit is PPAD-hard, and subsequent work has shown hardness results for other problems in PPAD by using -GCircuit as an intermediate problem. We introduce Pure-Circuit, a new intermediate problem for PPAD, which can be thought of as -GCircuit pushed to the limit as , and we show that the problem is PPAD-complete. We then prove that -GCircuit is PPAD-hard for all 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.
Paper Structure (84 sections, 35 theorems, 39 equations, 2 figures)

This paper contains 84 sections, 35 theorems, 39 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Our contribution.
  3. Open Questions.
  4. Proof Overview for Our Main Result
  5. The Pure-Circuit Problem
  6. The Generalized-Circuit problem.
  7. Taking the limit $\boldsymbol{\varepsilon\xspace \to 1}$.
  8. The PURIFY gate.
  9. Robustness of the logical gates.
  10. Formal definition.
  11. Alternative Gates and Further Restrictions
  12. More gates.
  13. More structure.
  14. PPAD-completeness of Pure-Circuit
  15. The $\textup{StrongSperner}$ Problem
  16. ...and 69 more sections

Key Result

Theorem 2.1

The Pure-Circuit problem is PPAD-complete.

Figures (2)

  • Figure 1: The gadgets used for win-lose polymatrix games.
  • Figure 2: The gadgets used for ThresholdGameNE.

Theorems & Definitions (69)

  • Definition 1: Pure-Circuit
  • Theorem 2.1
  • Remark 1
  • Corollary 2.2
  • proof
  • Corollary 2.3
  • Remark 2
  • Definition 2
  • Theorem 3.1: DaskalakisSZ21-min-max
  • Remark 3
  • ...and 59 more