Multiparty communication complexity

In theoretical computer science, multiparty communication complexity is the study of communication complexity in the setting where there are more than 2 players.

In the traditional two–party communication game, introduced by Yao (1979),^[1] two players, P₁ and P₂ attempt to compute a Boolean function

${\displaystyle f(x_{1},x_{2}):\{0,1\}^{n}\to \{0,1\},\ x_{1},x_{2}\in \{0,1\}^{n'},\ 2n'=n}$

Player P₁ knows the value of x₂, P₂ knows the value of x₁, but P_i does not know the value of x_i, for i = 1, 2.

In other words, the players know the other's variables, but not their own. The minimum number of bits that must be communicated by the players to compute f is the communication complexity of f, denoted by κ(f).

The multiparty communication game, defined in 1983,^[2] is a powerful generalization of the 2–party case: Here the players know all the others' input, except their own. Because of this property, sometimes this model is called "numbers on the forehead" model, since if the players were seated around a round table, each wearing their own input on the forehead, then every player would see all the others' input, except their own.

The formal definition is as follows: ${\displaystyle k}$ players: ${\displaystyle P_{1},P_{2},...,P_{k}}$ intend to compute a Boolean function

${\displaystyle f(x_{1},x_{2},\ldots ,x_{n}):\{0,1\}^{n}\to \{0,1\}}$

On set ${\displaystyle S=\{x_{1},x_{2},...,x_{n}\}}$ of variables there is a fixed partition ${\displaystyle A}$ of ${\displaystyle k}$ classes ${\displaystyle A_{1},A_{2},...,A_{k}}$, and player ${\displaystyle P_{1}}$ knows every variable, except those in ${\displaystyle A_{i}}$, for ${\displaystyle i=1,2,...,k}$. The players have unlimited computational power, and they communicate with the help of a blackboard, viewed by all players.

The aim is to compute ${\displaystyle f(x_{1},x_{2},...,x_{n}}$), such that at the end of the computation, every player knows this value. The cost of the computation is the number of bits written onto the blackboard for the given input ${\displaystyle x=(x_{1},x_{2},...,x_{n})}$ and partition ${\displaystyle A=(A_{1},A_{2},...,Ax_{n})}$. The cost of a multiparty protocol is the maximum number of bits communicated for any ${\displaystyle x}$ from the set {0,1}ⁿ and the given partition ${\displaystyle A}$. The ${\displaystyle k}$-party communication complexity, ${\displaystyle C_{A}^{(k)}(f)}$ of a function ${\displaystyle f}$, with respect to partition ${\displaystyle A}$, is the minimum of costs of those ${\displaystyle k}$-party protocols which compute ${\displaystyle f}$. The ${\displaystyle k}$-party symmetric communication complexity of ${\displaystyle f}$ is defined as

${\displaystyle C^{(k)}(f)=\max _{A}C_{A}^{(k)}(f)}$

where the maximum is taken over all k-partitions of set ${\displaystyle x=(x_{1},x_{2},...,x_{n})}$.