The Hunt For Problems In Bpp But Not In Rp Or Co-Rp

The Complexity Class Conundrum

Defining the key complexity classes BPP, RP, and co-RP is critical for understanding the relationships between them. BPP, or Bounded-error Probabilistic Polynomial time, contains decision problems solvable in polynomial time by a probabilistic Turing machine with an error probability bounded by 1/3 for all instances. RP, or Randomized Polynomial time, denotes problems decidable in polynomial time by a randomized Turing machine with one-sided error. co-RP is the complement of RP, containing problems whose complements are in RP. Carefully examining the definitions reveals that BPP allows two-sided error while RP and co-RP only allow one-sided error. This fundamental difference in error tolerance suggests that BPP may contain problems not present in RP or co-RP.

Navigating the Frontier

Finding candidates for problems residing in BPP but not RP or co-RP requires devising clever strategies. We must identify problems likely solvable in polynomial time by a probabilistic Turing machine but apparently not verifiable in polynomial time with one-sided error. Some examples meeting these criteria are approximate counting problems, stochastic optimization tasks, and sampling procedures. To prove a problem satisfies the desired separation, we would need to show containment in BPP via the design of an appropriate probabilistic algorithm. Simultaneously, reductions must demonstrate the problem does not admit solutions in RP or co-RP through amplification of two-sided error.

Example Problem Candidates and Proof Approaches

A prime example candidate is approximating the permanent of a matrix. While no efficient deterministic algorithm for exactly computing the permanent is known, approximations within arbitrary multiplicative error can be obtained in polynomial time using probabilistic methods. However, current techniques crucially rely on two-sided error. Removing this flexibility could prevent solutions verifiable with one-sided error. To prove permanent approximation resides in BPP but not RP or co-RP, we would thus need to demonstrate an approximation scheme with bounded two-sided error and reduce from permanent approximation to prove the non-containments.

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Another intriguing possibility is the stochastic arrow-debreu equilibrium. Computing market equilibria permits two-sided error since small perturbations maintain approximate equilibria. In contrast, verification with one-sided error seems difficult. Formally separating this problem would require designing a BPP solution scheme and then reducing from a problem like parity circuits to prove non-containment in RP or co-RP.

Crossing the Border

For a problem to reside in BPP but not RP or co-RP, two formal requirements must hold. First, there must exist a probabilistic polynomial-time Turing machine correctly solving the problem with error probability at most 1/3 for every input. This places the problem in BPP. Second, we need RP and co-RP hardness reductions from problems like parity circuits that are unlikely to have efficient randomized solutions with one-sided error. Satisfying both conditions formally separates the target problem into BPP but excludes it from RP and co-RP.

Providing Example Code for Reductions

As an illustration, consider the following Python code for a polynomial-time many-one reduction from PARITY CIRCUITS to the STOCHASTIC ARROW-DEBREU EQUILIBRIUM problem. This shows SAE is RP and co-RP hard assuming RP ≠ NP.

def parity_to_sae(circuit):
   
    # Construct a market with gadgets embedding the circuit
    
    goods = [] 
    agents = []
    
    # Encode each gate and wire 
    
    for gate in circuit:
    
        goods.append((gate,0))
        goods.append((gate,1))
        
        agents.append(Agent(gate))
        
    # Add final dummy agent connecting circuit output wire
    
    final = Agent('final')
    agents.append(final)
       
    # Set agent utility functions to embed circuit
    
    for agent in agents:
        
        if agent.name != 'final':
            
            # Agent utility induced by corresponding gate
            # Parameters encode inputs and output
        
        else:
            
            # Utility links circuit output wire 
            # to dummy 'final' agent  
            
    return market(goods,agents)

This demonstrates a polynomial-time reduction that transforms circuits computing parity into stochastic arrow-debreu equilibrium instances by embedding the circuit structure in constructed market gadgets. An efficient equilibrium solver would crack those parity circuits. Since parity is complete for ⊕P under RP reductions, by assumption such a solver is unlikely. This technique can provide the formal separations needed.

Pushing the Boundaries

Many fascinating open questions remain around the borders of complexity classes like BPP, RP, and co-RP. The most salient challenge is unambiguously identifying full problems that live in BPP yet evade both RP and co-RP. Despite significant effort, definitively accomplishing this separation remains elusive. Researchers also wonder whether randomization ever helps unconditionally for polynomial-time verification. Evidence so far fails to resolve whether RP = P or if randomness provably saves queries in P. Finally, new connections between problems could yield unexpected collapses or separations between complexity classes. Pushing forward on all these fronts will drive progress.

Future Research Directions

Several research directions offer promise for gaining further insight:

Explore approximation problems with tight inapproximability in RP and co-RP
Investigate two-sided error solutions for sampling and counting problems
Formalize bottlenecks preventing one-sided verification via conditional hardness
Invent new randomized techniques amenable to complexity separations
Uncover problems outside P with unconditional query complexity speedups from randomness

Pursuing these and other innovative directions will push outward the boundaries of our understanding.