Class Description

BPP^cc: Communication Complexity BPP

The analogue of P^cc for bounded-error probabilistic communication complexity.

Does not equal P^cc, and is not contained in NP^cc, because of the EQUALITY problem.

If the classes are defined in terms of partial functions, then BPP^cc

• is not contained in P^NPcc [PPS14];
• does not contain NP^cc ∩ coNP^cc [Kla03].

Linked From

BPP_{${\displaystyle k}$}^cc: BPP^cc in NOF model, ${\displaystyle k}$ players

Has the same relation to BPP^cc and BPP as P_{${\displaystyle k}$}^cc does to P^cc and P.

NP_{${\displaystyle k}$}^cc is not contained in BPP_{${\displaystyle k}$}^cc for ${\displaystyle k\leq (1-\delta )\cdot \log n}$ players, for any constant ${\displaystyle \delta >0}$ [DP08].

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NP^cc: Communication Complexity NP

The analogue of P^cc for nondeterministic communication complexity. Both communication bits and nondeterministic guess bits count toward the complexity.

Does not equal P^cc or coNP^cc because of the EQUALITY problem. Also, does not contain BPP^cc because of that problem.

Is not contained in BPP^cc (first shown by [BFS86]).

SET-INTERSECTION, in which Alice's and Bob's strings are identified with characteristic vectors of sets and yes-inputs are intersecting while no-inputs are disjoint, is the canonical NP^cc-complete problem. Sometimes SET-DISJOINTNESS also refers to this problem, but sometimes it refers to the complement of this problem.

The complexity measure corresponding to NP^cc is equivalent to the log of the number of rectangles needed to cover exactly the set of 1-entries of the communication matrix.

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P^cc: Communication Complexity P

In a two-party communication complexity problem, Alice and Bob have n-bit strings x and y respectively, and they wish to evaluate some Boolean function f(x,y) using as few bits of communication as possible. P^cc is the class of (infinite families of) f's, such that the amount of communication needed is only O(polylog(n)), even if Alice and Bob are restricted to a deterministic protocol.

Note that all functions of the form above are solvable given ${\displaystyle O(n)}$ bits of communication, since no bounds are placed on the computational abilities of Alice and Bob. Thus, when discussing this class, "polynomially" is sometimes used in place of "polylogarithmically."

Is strictly contained in NP^cc and in BPP^cc because of the EQUALITY problem.

If the classes are defined in terms of total functions, then P^cc = NP^cc ∩ coNP^cc = UP^cc.

Defined in [BFS86].

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PH^cc: Communication Complexity PH

The polynomial hierarchy for communication complexity, naturally built with NP^cc and coNP^cc forming the first level. Specifically, a cost-k protocol in the PH^cc model corresponds to a constant-depth 2^k-ary tree whose internal nodes are alternating layers of AND and OR gates, and whose leaves represent (the indicator functions of) either rectangles or complements of rectangles, as appropriate.

It is unknown whether Σ₂^cc equals Π₂^cc.

Defined in [BFS86], where it was also shown (among other things) that BPP^cc is contained in Σ₂^cc ∩ Π₂^cc.

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P^NPcc: Communication Complexity P^NP

Not contained in PP^cc [BVW07].

Does not contain BPP^cc if partial functions are allowed [PPS14].

Contained in UPostBPP^cc.

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