Special Topics in Complexity Theory, Lecture 15

Special Topics in Complexity Theory, Fall 2017. Instructor: Emanuele Viola

1 Lecture 15, Scribe: Chin Ho Lee

In this lecture fragment we discuss multiparty communication complexity, especially the problem of separating deterministic and randomized communication, which we connect to a problem in combinatorics.

2 Number-on-forehead communication complexity

In number-on-forehead (NOH) communication complexity each party i sees all of the input (x_1, \dotsc , x_k) except its own input x_i. For background, it is not known how to prove negative results for k \ge \log n parties. We shall focus on the problem of separating deterministic and randomizes communication. For k = 2, we know the optimal separation: The equality function requires \Omega (n) communication for deterministic protocols, but can be solved using O(1) communication if we allow the protocols to use public coins. For k = 3, the best known separation between deterministic and randomized protocol is \Omega (\log n) vs O(1) [BDPW10]. In the following we give a new proof of this result, for a simpler function: f(x, y, z) = 1 if and only if x \cdot y \cdot z = 1 for x, y, z \in SL_2(q).

For context, let us state and prove the upper bound for randomized communication.

Claim 1. f has randomized communication complexity O(1).

Proof. In the NOH model, computing f reduces to 2-party equality with no additional communication: Alice computes y \cdot z =: w privately, then Alice and Bob check if x = w^{-1}. \square

To prove a \Omega (\log n) lower bound for deterministic protocols, where n = \log |G|, we reduce the communication problem to a combinatorial problem.

Definition 2. A corner in a group G is \{ (x,y), (xz, y), (x,zy) \} \subseteq G^2, where x, y are arbitrary group elements and z \neq 1_G.

For intuition, consider the case when G is Abelian, where one can replace multiplication by addition and a corner becomes \{ (x, y), (x + z, y), (x, y + z)\} for z \neq 0.

We now state the theorem that gives the lower bound.

Theorem 3. Suppose that every subset A \subseteq G^2 with \mu (A) := |A|/|G^2| \ge \delta contains a corner. Then the deterministic communication complexity of f(x, y, z) = 1 \iff x \cdot y \cdot z = 1_G is \Omega (\log (1/\delta )).

It is known that when G is Abelian, then \delta \ge 1/\mathrm {polyloglog}|G| implies a corner. We shall prove that when G = SL_2(q), then \delta \ge 1/\mathrm {polylog}|G| implies a corner. This in turn implies communication \Omega (\log \log |G|) = \Omega (\log n).

Proof. We saw that a number-in-hand (NIH) c-bit protocol can be written as a disjoint union of 2^c rectangles. Likewise, a number-on-forehead c-bit protocol P can be written as a disjoint union of 2^c cylinder intersections C_i := \{ (x, y, z) : f_i(y,z) g_i(x,z) h_i(x,y) = 1\} for some f_i, g_i, h_i\colon G^2 \to \{0, 1\}:

\begin{aligned} P(x,y,z) = \sum _{i=1}^{2^c} f_i(y,z) g_i(x,z) h_i(x,y). \end{aligned}

The proof idea of the above fact is to consider the 2^c transcripts of P, then one can see that the inputs giving a fixed transcript are a cylinder intersection.

Let P be a c-bit protocol. Consider the inputs \{(x, y, (xy)^{-1}) \} on which P accepts. Note that at least 2^{-c} fraction of them are accepted by some cylinder intersection C. Let A := \{ (x,y) : (x, y, (xy)^{-1}) \in C \} \subseteq G^2. Since the first two elements in the tuple determine the last, we have \mu (A) \ge 2^{-c}.

Now suppose A contains a corner \{ (x, y), (xz, y), (x, zy) \}. Then

\begin{aligned} (x,y) \in A &\implies (x, y, (xy)^{-1}) \in C &&\implies h(x, y) = 1 , \\ (xz,y) \in A &\implies (xz, y, (xzy)^{-1}) \in C &&\implies f(y,(xyz)^{-1}) = 1 , \\ (x,zy) \in A &\implies (x, zy, (xzy)^{-1}) \in C &&\implies g(x,(xyz)^{-1}) = 1 . \end{aligned}

This implies (x,y,(xzy)^{-1}) \in C, which is a contradiction because z \neq 1 and so x \cdot y \cdot (xzy)^{-1} \neq 1_G. \square

References

[BDPW10]   Paul Beame, Matei David, Toniann Pitassi, and Philipp Woelfel. Separating deterministic from randomized multiparty communication complexity. Theory of Computing, 6(1):201–225, 2010.

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