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Half-duplex communication complexity with adversary can be less than the classical communication complexity

Mikhail Dektiarev, Nikolay Vereshchagin

TL;DR

The paper investigates half-duplex communication complexity with adversaries and proves a clean separation from classical communication complexity: there exists a total function whose half-duplex complexity with a malicious adversary is strictly smaller than its classical complexity (5 vs 6). It also provides a partial-function with a logarithmic gap between local half-duplex/classical complexity and demonstrates a constant-gap separation in the honest-adversary setting. The authors introduce novel techniques based on fooling-rectangle expansions and graph-expansion arguments to derive lower bounds, and they construct explicit protocol realizations (U and M) whose matrices drive these separations. These results illuminate intrinsic differences between half-duplex and classical models and motivate several open questions about gaps in both partial and total function settings.

Abstract

Half-duplex communication complexity with adversary was defined in [Hoover, K., Impagliazzo, R., Mihajlin, I., Smal, A. V. Half-Duplex Communication Complexity, ISAAC 2018.] Half-duplex communication protocols generalize classical protocols defined by Andrew Yao in [Yao, A. C.-C. Some Complexity Questions Related to Distributive Computing (Preliminary Report), STOC 1979]. It has been unknown so far whether the communication complexities defined by these models are different or not. In the present paper we answer this question: we exhibit a function whose half-duplex communication complexity with adversary is strictly less than its classical communication complexity.

Half-duplex communication complexity with adversary can be less than the classical communication complexity

TL;DR

The paper investigates half-duplex communication complexity with adversaries and proves a clean separation from classical communication complexity: there exists a total function whose half-duplex complexity with a malicious adversary is strictly smaller than its classical complexity (5 vs 6). It also provides a partial-function with a logarithmic gap between local half-duplex/classical complexity and demonstrates a constant-gap separation in the honest-adversary setting. The authors introduce novel techniques based on fooling-rectangle expansions and graph-expansion arguments to derive lower bounds, and they construct explicit protocol realizations (U and M) whose matrices drive these separations. These results illuminate intrinsic differences between half-duplex and classical models and motivate several open questions about gaps in both partial and total function settings.

Abstract

Half-duplex communication complexity with adversary was defined in [Hoover, K., Impagliazzo, R., Mihajlin, I., Smal, A. V. Half-Duplex Communication Complexity, ISAAC 2018.] Half-duplex communication protocols generalize classical protocols defined by Andrew Yao in [Yao, A. C.-C. Some Complexity Questions Related to Distributive Computing (Preliminary Report), STOC 1979]. It has been unknown so far whether the communication complexities defined by these models are different or not. In the present paper we answer this question: we exhibit a function whose half-duplex communication complexity with adversary is strictly less than its classical communication complexity.
Paper Structure (23 sections, 17 theorems, 38 equations, 13 figures, 1 table)

This paper contains 23 sections, 17 theorems, 38 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A separation of half-duplex and classical communication complexities for partial functions
  4. An example
  5. Local communication complexity
  6. An example
  7. Local and global complexity of total functions
  8. Local classical complexity of the function $g$
  9. A separation of half-duplex complexity with honest adversary from classical complexity for total functions
  10. A separation of half-duplex complexity with malicious adversary from classical complexity for total functions
  11. The half-duplex protocol
  12. A general protocol
  13. The first realization of the plan
  14. The second (and final) realization of the plan
  15. Sub-functions of $U$ with communication complexity 6: the function $S$
  16. ...and 8 more sections

Key Result

Lemma 1

There is a partial function $g$ whose half-duplex complexity is $1$ and classical complexity is at least $2$.

Figures (13)

  • Figure 1: A protocol of depth 3 to compute a function $f:\{0,1\}^3\times \{0,1\}^3\to\{1,2,3,4\}$. Each internal node is labeled by a letter indicating the turn to move and by a function computing the bit to send. For instance, if Alice has the string $x=010$ and Bob the string $y=110$, then in the first round Alice sends 1, in the second round Alice sends 0 and in the third round Bob sends 1. Then both parties output 4.
  • Figure 2: A 1-round half-duplex protocol. Here we assume that $X=Y=\{0,1,2\}$, $Z=\{0,1\}$. The event "sent $i$" is abbreviated as s$i$ and similarly the event "receive $i$" is abbreviated as r$i$. The letter $g$ denotes the mapping $0\mapsto \text{send 0}, 1\mapsto \text{send 1}, 2\mapsto \text{receive}$, and $h$ denotes the same mapping. This protocol computes the function $f:\{0,1,2\}\times \{0,1,2\}\to\{0,1\}$ which is defined only on the pairs $(0,0), (1,1), (1,2)$. On the first and the second pair both players send a bit which is lost, however they output the same result. On the third pair Alice sends 1, which is received by Bob, who actually does not care, since he outputs 1 any way.
  • Figure 3: The part of matrix of function $g$ consisting of simple inputs.
  • Figure 4: The matrix of the function $S$. Its classical complexity is 6 and its half-duplex complexity is at most 5.
  • Figure 5: The matrix of the function $M$. Its classical communication complexity is 6, while its half-duplex complexity is at most 5.
  • ...and 8 more figures

Theorems & Definitions (42)

  • Definition 1: yao
  • Definition 2
  • Definition 3
  • Definition 4: hims
  • Definition 5
  • Definition 6
  • Remark 1
  • Lemma 1
  • proof
  • Theorem 1
  • ...and 32 more