Analytical Approach to Parallel Repetition (1305.1979v3)

Abstract: We propose an analytical framework for studying parallel repetition, a basic product operation for one-round two-player games. In this framework, we consider a relaxation of the value of a game, $\mathrm{val}+$, and prove that for projection games, it is both multiplicative (under parallel repetition) and a good approximation for the true value. These two properties imply a parallel repetition bound as $$ \mathrm{val}(G^{\otimes k}) \approx \mathrm{val}+(G^{\otimes k}) = \mathrm{val}_+(G)^{k} \approx \mathrm{val}(G)^{k}. $$ Using this framework, we can also give a short proof for the NP-hardness of Label-Cover$(1,\delta)$ for all $\delta>0$, starting from the basic PCP theorem. We prove the following new results: - A parallel repetition bound for projection games with small soundness. Previously, it was not known whether parallel repetition decreases the value of such games. This result implies stronger inapproximability bounds for Set-Cover and Label-Cover. - An improved bound for few parallel repetitions of projection games, showing that Raz's counterexample is tight even for a small number of repetitions. Our techniques also allow us to bound the value of the direct product of multiple games, namely, a bound on $\mathrm{val}(G_1\otimes ...\otimes G_k)$ for different projection games $G_1,...,G_k$.

• The paper introduces a relaxed game value that is proven to be multiplicative over parallel repetitions, providing improved bounds for projection games.
• It demonstrates that even sub-constant game values decrease significantly with repetition, countering previous assumptions and linking to NP-hardness in approximation.
• The framework extends Raz's counterexample results, offering a uniform analytical method for understanding parallel repetition behavior across varying repetition numbers.

An Analytical Approach to Parallel Repetition in Projection Games

In this paper, the researchers propose an analytical framework for studying parallel repetition in one-round two-player games, specifically focusing on projection games. The framework provides significant insights into the multiplicative properties of parallel repetition, aiming to advance our understanding of how game values change under repetition.

Overview of Key Contributions

The primary contribution is a novel analytical technique for assessing the parallel repetition of projection games. The authors introduce a relaxed game value that is shown to be multiplicative across repetitions, leading to improved bounds on game values. This relaxation addresses long-standing questions about whether parallel repetition necessarily decreases the value of games, particularly those with already low values.

Results and Implications

• Improved Parallel Repetition Bound: For a given projection game $G$ with value at most $\rho$, its $k$-fold parallel repetition has a bound:

$$\mathrm{val}(G^{\otimes k}) \leq \left(\frac{2\sqrt{\rho}}{1+\rho}\right)^{k/2}$$

This provides clear evidence that even sub-constant values of $G$ are driven lower through repetition, contradicting previous assumptions.

• NP-hardness Results: The paper establishes that approximating set cover to within a factor of $(1-\epsilon) \ln n$ is NP-hard for any $\epsilon > 0$. This strengthens previous results and connects parallel repetition with foundational problems in theoretical computer science.
• Tight Bounds for Few Repetitions: For small numbers of repetitions, the paper shows that Raz's counterexample to strong parallel repetition is tight, extending known results and providing a uniform explanation of parallel repetition behavior as the number of repetitions varies.

Practical and Theoretical Implications

The implications of this work extend into both practical applications in computational complexity and theoretical explorations of game repetition properties:

• Hardness of Approximation: The insights into parallel repetition contribute to a broader understanding of complexity theory, influencing reductions used in proofs of the NP-hardness of approximation problems.
• Potential for Future Research: As a foundation, this framework invites further investigation into different types of games, such as entangled games or games with more than two players, prompting new research directions beyond classical projection games.

Future Developments and Directions

The results presented here may encourage researchers to explore generalized approaches to parallel repetition across other game types. The application of this analytical method to entangled game frameworks could yield similarly potent results, especially given the challenges inherent in quantum game theory.

Overall, this paper contributes a robust analytical toolset for examining parallel repetition, challenging existing hypotheses about game theory and opening avenues for richer computational complexity research. The authors have effectively leveraged operator norms and copositive programming within their framework, setting a precedent for future exploration in this area.