• Corpus ID: 14918700

On Fortification of General Games

  title={On Fortification of General Games},
  author={Amey Bhangale and Ramprasad Saptharishi and G. Varma and Rakesh Venkat},
  journal={Electron. Colloquium Comput. Complex.},
A recent result of Moshkovitz [Mos14] presented an ingenious method to provide a completely elementary proof of the Parallel Repetition Theorem for certain projection games via a construction called fortification. However, the construction used in [Mos14] to fortify arbitrary label cover instances using an arbitrary extractor is insufficient to prove parallel repetition. In this paper, we provide a fix by using a stronger graph that we call fortifiers. Fortifiers are graphs that have both ‘1… 

Figures from this paper



Parallel Repetition from Fortification

  • Dana Moshkovitz
  • Mathematics, Computer Science
    2014 IEEE 55th Annual Symposium on Foundations of Computer Science
  • 2014
This work gives a simple transformation on games -- "fortification" -- and shows that for fortified games, the value of the repeated game decreases perfectly exponentially with the number of repetitions, up to an arbitrarily small additive error.

Analytical approach to parallel repetition

Improved bounds for few parallel repetitions of projection games are shown, showing that Raz's counterexample to strong parallel repetition is tight even for a small number of repetitions, and a short proof for the NP-hardness of label cover(1, δ) for all δ > 0, starting from the basic PCP theorem.

Small Value Parallel Repetition for General Games

A parallel repetition theorem for general games with value tending to 0 is proved and the small-value parallel repetition bound obtained is tight.

A Counterexample to Strong Parallel Repetition

  • R. Raz
  • Mathematics
    2008 49th Annual IEEE Symposium on Foundations of Computer Science
  • 2008
A major motivation for the recent interest in the strong parallel repetition problem is that a strong Parallel repetition theorem would have implied that the unique game conjecture is equivalent to the NP hardness of distinguishing between instances of Max-Cut that are at least 1 - isin2 satisfiable from instances that areat most 1 - (2/pi) ldr isin satisfiable.

Lifts, Discrepancy and Nearly Optimal Spectral Gap*

It is shown that every graph of maximal degree d has a 2-lift such that all “new” eigenvalues are in the range, leading to a deterministic polynomial time algorithm for constructing arbitrarily large d-regular graphs, with second eigenvalue O(d/α)+1.


We present a new explicit construction for expander graphs with nearly optimal spectral gap. The construction is based on a series of 2-lift operations. Let G be a graph on n vertices. A 2-lift of G

Complexity-Theoretic Aspects of Interactive Proof Systems

This thesis will show that for any language that has a perfect zero-knowledge proof system, its complement has a short interactive protocol, which implies that there are not any perfectzero-knowledge protocols for NP-complete languages unless the polynomial-time hierarchy collapses.

A parallel repetition theorem

  • R. Raz
  • Computer Science, Mathematics
    STOC '95
  • 1995
We show that a parallel repetition of any two-prover one-round proof system (MIP(2,1)) decreases the probability of error at an exponential rate. No constructive bound was previously known. The

Parallel repetition in projection games and a concentration bound

This work improves earlier results of Raz and Holenstein and proves the bound (1-ε2)Ω(n) as long as the game is a "projection game", the type of game most commonly used in hardness of approximation results, and proves a concentration bound for parallel repetition.

Two-Prover Protocols - Low Error at Affordable Rates

It is shown that for any $\epsilon >0,$ NP has two-prover one-round proof systems with logarithmic-sized questions, constant-sized answers, and error at most $\ep silon$.