Parallel Repetition Theorem

Definition

The parallel repetition theorem is a concept in computational complexity theory that states that repeating a game multiple times in parallel increases the difficulty of approximating its value. Specifically, if a game can be approximated with some accuracy, repeating the game independently many times leads to a drastically lower probability of success for any strategy that tries to approximate the outcome. This theorem is particularly relevant in understanding hardness of approximation results and sheds light on the computational power of interactive proofs.

5 Must Know Facts For Your Next Test

1. The parallel repetition theorem provides a rigorous framework that shows how the complexity of approximating a game's value changes with repeated trials.
2. When games are repeated independently, the success probability for any approximating strategy diminishes exponentially with the number of repetitions.
3. This theorem applies primarily to two-player games and has significant implications for hardness results in computational complexity.
4. It can be used to derive strong inapproximability results for various NP-hard problems by demonstrating that their approximability weakens when considered under parallel repetitions.
5. The parallel repetition theorem indicates that certain interactive proof systems gain strength when repeated, thereby influencing their effectiveness in verifying complex statements.

Review Questions

• How does the parallel repetition theorem impact the ability to approximate the value of games?
  • The parallel repetition theorem suggests that as games are repeated in parallel, the likelihood of successfully approximating the game's value significantly decreases. For instance, if a game has a certain approximation ratio, repeating it multiple times independently leads to an exponential drop in success probability for any strategy. This means that strategies which might work well for a single instance become ineffective when applied across multiple instances.
• Discuss how the parallel repetition theorem relates to hardness of approximation results for NP-hard problems.
  • The parallel repetition theorem serves as a foundational tool in establishing hardness of approximation results for NP-hard problems. By showing that repeating instances of a problem lowers the chances of achieving good approximations, researchers can conclude that if a problem is easy to approximate once, it becomes increasingly hard when repeated. This connection allows for stronger inapproximability results across various optimization problems.
• Evaluate the implications of the parallel repetition theorem on interactive proof systems and their computational power.
  • The implications of the parallel repetition theorem on interactive proof systems are profound. When these systems are repeated, they often exhibit enhanced verification strength due to increased communication and reduced probabilities of error. This means that repeated interactive proofs can provide more confidence in their correctness, making them powerful tools in complexity theory and cryptography. Such insights not only improve our understanding of these systems but also pave the way for new applications in secure computation.

Related terms

Hardness of Approximation: A field in computational complexity that studies the difficulty of finding approximate solutions to optimization problems, particularly when exact solutions are computationally infeasible.

Interactive Proof Systems: A type of proof system where a prover sends messages to a verifier, who then decides whether to accept or reject based on the messages received, highlighting the role of communication in computational problems.

Approximation Ratio: A measure used to evaluate how close an approximate solution is to the optimal solution, often expressed as a ratio of the two values.