We study the CHAIN communication problem introduced by Cormode et al. [ICALP 2019]. It is a generalization of the well-studied INDEX problem. For $k\geq 1$, in CHAIN$_{n,k}$, there are $k$ instances of INDEX, all with the same answer. They are shared between $k+1$ players as follows. Player 1 has the first string $X^1 \in \{0,1\}^n$, player 2 has the first index $\sigma^1 \in [n]$ and the second string $X^2 \in \{0,1\}^n$, player 3 has the second index $\sigma^2 \in [n]$ along with the third string $X^3 \in \{0,1\}^n$, and so on. Player $k+1$ has the last index $\sigma^k \in [n]$. The communication is one way from each player to the next, starting from player 1 to player 2, then from player 2 to player 3 and so on. Player $k+1$, after receiving the message from player $k$, has to output a single bit which is the answer to all $k$ instances of INDEX. It was proved that the CHAIN$_{n,k}$ problem requires $\Omega(n/k^2)$ communication by Cormode et al., and they used it to prove streaming lower bounds for approximation of maximum independent sets. Subsequently, it was used by Feldman et al. [STOC 2020] to prove lower bounds for streaming submodular maximization. However, these works do not get optimal bounds on the communication complexity of CHAIN$_{n,k}$, and in fact, it was conjectured by Cormode et al. that $\Omega(n)$ bits are necessary, for any $k$. As our main result, we prove the optimal lower bound of $\Omega(n)$ for CHAIN$_{n,k}$. This settles the open conjecture of Cormode et al. in the affirmative. The key technique is to use information theoretic tools to analyze protocols over the Jensen-Shannon divergence measure, as opposed to total variation distance. As a corollary, we get an improved lower bound for approximation of maximum independent set in vertex arrival streams through a reduction from CHAIN directly.

翻译：我们研究Cormode等人[ICALP 2019]提出的CHAIN通信问题，该问题是经典INDEX问题的推广。对于$k\geq 1$，CHAIN$_{n,k}$包含$k$个共享相同答案的INDEX实例，这些实例分布在$k+1$个玩家之间：玩家1持有首个字符串$X^1 \in \{0,1\}^n$，玩家2持有首个索引$\sigma^1 \in [n]$及第二个字符串$X^2 \in \{0,1\}^n$，玩家3持有第二个索引$\sigma^2 \in [n]$及第三个字符串$X^3 \in \{0,1\}^n$，依此类推。玩家$k+1$持有最后一个索引$\sigma^k \in [n]$。通信采用单向传递模式：从玩家1到玩家2，再从玩家2到玩家3，依次进行。玩家$k+1$在收到来自玩家$k$的消息后需输出单个比特，该比特应为所有$k$个INDEX实例的答案。Cormode等人曾证明CHAIN$_{n,k}$问题需要$\Omega(n/k^2)$通信量，并利用该结果推导了最大独立集近似问题的流式下界。随后Feldman等人[STOC 2020]将该结果用于证明流式子模最大化问题的下界。然而这些工作未获得CHAIN$_{n,k}$通信复杂度的最优界，Cormode等人甚至猜想对于任意$k$，$\Omega(n)$比特是必要的。作为本文主要成果，我们证明了CHAIN$_{n,k}$的最优下界$\Omega(n)$，从而肯定解决了Cormode等提出的开放猜想。关键技术在于使用信息论工具分析基于Jensen-Shannon散度（而非总变差距离）的协议。作为推论，通过CHAIN的直接归约，我们获得了顶点到达流中最大独立集近似问题的改进下界。