Derandomizing The Valiant-Vazirani Theorem: Possibilities And Limitations
Derandomizing Valiant-Vazirani: The Core Problem
The Valiant-Vazirani theorem shows that any Boolean formula can be made satisfiable by a random assignment of truth values to its variables with high probability. However, generating truly random bits requires specialized hardware and can be inefficient. Derandomization aims to reduce or eliminate the need for randomness while preserving the power of randomized algorithms.
The core problem is developing deterministic algorithms that can simulate randomized algorithms without needing access to a source of random bits. This involves finding alternative sources of unpredictability to substitute for randomness. If successful, derandomization would allow randomized algorithms to run efficiently on conventional computers.
Reducing Randomness in Boolean Formula Satisfiability
Boolean formula satisfiability is the problem of finding an assignment of true/false values to variables to make the whole formula evaluate to true. The Cook-Levin theorem showed this problem encapsulates the intrinsic difficulty of general computation. Surprisingly, Valiant and Vazirani proved that for any Boolean formula, if truth values are assigned randomly, the formula will be satisfied with high probability.
However, generating random bits incurs computational costs and physical limitations. Reducing or eliminating the need for randomness could lead to more efficient satisfiability algorithms. But deterministic algorithms cannot guarantee satisfiability for all formulas. Bridging this gap is an active area of research.
The Valiant-Vazirani Theorem on Randomness Reduction
The seminal Valiant-Vazirani theorem states that for a Boolean formula with $n$ variables and size $m$, choosing truth values for each variable uniformly at random will satisfy the formula with probability at least $1 – \frac{m}{2^n}$. So if $m << 2^n$, a random assignment succeeds in satisfying the formula with high probability.
This theorem initiated the field of derandomization, which aims to efficiently generate randomness or simulate randomized algorithms without physical sources of randomness. Success would allow randomized algorithms to be run deterministically on conventional computers. However, current derandomization techniques still fall short of fully bridging the gap.
Techniques for Derandomization
Two main approaches have been studied for derandomizing algorithms:
Randomness Extractors
A randomness extractor converts a weakly random source into a higher quality source of randomness for use in randomized algorithms. Extractors require some source of initial randomness or unpredictability, known as the seed. High-quality output randomness can be generated from an imperfect input source. Extractors harness intrinsic randomness in the physical world.
Pseudorandom Generators
Pseudorandom generators (PRGs) deterministically generate randomness from a short truly random seed. The output appears random statistically despite being generated deterministically. PRGs can expand a small number of random bits into a longer stream. However, the unpredictability of PRG output is limited against adversaries with additional information.
Limitations of Current Derandomization Methods
While randomness extractors and PRGs reduce the amount of randomness needed by algorithms, some key limitations remain:
- Extractors still require some source of initial randomness to seed the extraction process.
- PRGs rely on a truly random seed and cannot output more randomness than contained in the seed.
- The output randomness may not withstand statistical tests or computationally bounded adversaries.
- There are still gaps between the capabilities of randomized and derandomized algorithms for problems such as unsatisfiability or lower bounds on circuit size.
Bridging these gaps remains an active area of research. More powerful techniques would be needed to fully replace randomness and match the capabilities of randomized algorithms.
Open Problems in Derandomization Research
Many open questions remain about the possibilities and limitations of derandomization:
- Can deterministic algorithms for Boolean formula satisfiability and other NP-complete problems match the performance of their randomized counterparts?
- Can derandomized circuit lower bound proofs match the tightness of bounds proven using randomness?
- Are there cryptography-secure PRGs that can withstand computationally bounded adversaries?
- Can physical sources of randomness like quantum measurements enable new derandomization breakthroughs?
As derandomization techniques mature, randomized algorithms may become far more practical on conventional computers. But eliminating randomness entirely remains an elusive goal for now. Ongoing research seeks to push the frontiers of unpredictability further, bringing the capabilities of deterministic computation closer to that of randomized computation.
