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Multiparty equality in the local broadcast model

Louis Esperet, Jean-Florent Raymond

TL;DR

This work analyzes multiparty equality in the local broadcast model through graph topology. It introduces a novel 2-connected-graph protocol, with cost bounded by half the size of a total vertex cover, and shows this yields near-optimal results for cycles, hypercubes, grids, and many regular graphs. By leveraging a faithful-host construction and special copies, the new protocol exploits total vertex covers to achieve substantial improvements over prior 4-approx methods in a broad set of topologies. The paper also develops and leverages linear-programming lower bounds for boundaries, and introduces a monotone variant of total vertex covers, outlining promising directions for extending these topology-aware techniques and posing open questions on optimality gaps and broader graph classes.

Abstract

In this paper we consider the multiparty equality problem in graphs, where every vertex of a graph $G$ is given an input, and the goal of the vertices is to decide whether all inputs are equal. We study this problem in the local broadcast model, where a message sent by a vertex is received by all its neighbors and the total cost of a protocol is the sum of the lengths of the messages sent by the vertices. This setting was studied by Khan and Vaidya, who gave in 2021 a protocol achieving a 4-approximation in the general case. We study this multiparty communication problem through the lens of network topology. We design a new protocol for 2-connected graphs, whose efficiency relies on the notion of total vertex cover in graph theory. This protocol outperforms the aforementioned 4-approximation in a number of cases. To demonstrate its applicability, we apply it to obtain optimal or asymptotically optimal protocols for several natural network topologies such as cycles, hypercubes, and grids. On the way we also provide new bounds of independent interest on the size of total vertex covers in regular graphs.

Multiparty equality in the local broadcast model

TL;DR

This work analyzes multiparty equality in the local broadcast model through graph topology. It introduces a novel 2-connected-graph protocol, with cost bounded by half the size of a total vertex cover, and shows this yields near-optimal results for cycles, hypercubes, grids, and many regular graphs. By leveraging a faithful-host construction and special copies, the new protocol exploits total vertex covers to achieve substantial improvements over prior 4-approx methods in a broad set of topologies. The paper also develops and leverages linear-programming lower bounds for boundaries, and introduces a monotone variant of total vertex covers, outlining promising directions for extending these topology-aware techniques and posing open questions on optimality gaps and broader graph classes.

Abstract

In this paper we consider the multiparty equality problem in graphs, where every vertex of a graph is given an input, and the goal of the vertices is to decide whether all inputs are equal. We study this problem in the local broadcast model, where a message sent by a vertex is received by all its neighbors and the total cost of a protocol is the sum of the lengths of the messages sent by the vertices. This setting was studied by Khan and Vaidya, who gave in 2021 a protocol achieving a 4-approximation in the general case. We study this multiparty communication problem through the lens of network topology. We design a new protocol for 2-connected graphs, whose efficiency relies on the notion of total vertex cover in graph theory. This protocol outperforms the aforementioned 4-approximation in a number of cases. To demonstrate its applicability, we apply it to obtain optimal or asymptotically optimal protocols for several natural network topologies such as cycles, hypercubes, and grids. On the way we also provide new bounds of independent interest on the size of total vertex covers in regular graphs.

Paper Structure

This paper contains 16 sections, 27 theorems, 12 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. A linear programming lower bound
  5. Simple protocols
  6. Total vertex covers and a new protocol
  7. The new protocol
  8. Applications
  9. Complete bipartite graphs $K_{2,t}$
  10. Cycles
  11. Hypercubes
  12. Grids
  13. Regular graphs
  14. Conclusion
  15. Monotone total vertex cover
  16. ...and 1 more sections

Key Result

Theorem 1.1

A simple protocol solves the multiparty equality problem if and only if the set $S \subseteq V$ of vertices chosen to transmit their entire input is a weakly connected dominating set of $G$. In particular, $\mathop{\mathrm{\mathsf{OPT}}}\nolimits(G)\le \mathsf{wds}(G)$.

Figures (2)

  • Figure 1: The vertex set $S$ (depicted as white circles) and the spanning 2-connected subgraph $G_S$ of the square grid, in red.
  • Figure 2: Total vertex covers (in white) of 2-connected spanning subgraphs (in red) of the triangular grid (left) and the grid with all diagonal (right).

Theorems & Definitions (40)

  • Theorem 1.1: khan2021testing
  • Theorem 1.2: khan2021testing
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Theorem 2.1: khan2021testing
  • Lemma 2.2
  • ...and 30 more