Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Improved Approximation for Broadcasting in k-cycle Graphs

Jeffrey Bringolf, Anne-Laure Ehresmann, Hovhannes A. Harutyunyan

TL;DR

This work studies the broadcast time problem on $k$-cycle graphs, a close relative of cactus graphs, where a message spreads in discrete rounds with at most one call per informed vertex per round. The authors introduce Simple-K-Cycle, a simple linear-time algorithm that handles two originator scenarios (central vertex vs. a cycle vertex at distance $d$) and fixes a predetermined call order to achieve a $1.5-\\epsilon$-approximation. They prove the approximation bound by case analysis, deriving per-cycle completion times and comparing to lower bounds on the optimal broadcast time, thereby improving the prior best known ratio of 2. The results advance understanding of broadcast time on near-cactus structures and suggest directions for future work, including exact complexity status and potential extensions to broader cactus families or PTAS results.

Abstract

Broadcasting is an information dissemination primitive where a message originates at a node (called the originator) and is passed to all other nodes in the network. Broadcasting research is motivated by efficient network design and determining the broadcast times of standard network topologies. Verifying the broadcast time of a node $v$ in an arbitrary network $G$ is known to be NP-hard. Additionally, recent findings show that the broadcast time problem is also NP-complete in general cactus graphs and some highly restricted subfamilies of cactus graphs. These graph families are structurally similar to $k$-cycle graphs, in which the broadcast time problem is also believed to be NP-complete. In this paper, we present a simple $(1.5-ε)$-approximation algorithm for determining the broadcast time of networks modeled using $k$-cycle graphs, where $ε> 0$ depends on the structure of the graph.

Improved Approximation for Broadcasting in k-cycle Graphs

TL;DR

This work studies the broadcast time problem on -cycle graphs, a close relative of cactus graphs, where a message spreads in discrete rounds with at most one call per informed vertex per round. The authors introduce Simple-K-Cycle, a simple linear-time algorithm that handles two originator scenarios (central vertex vs. a cycle vertex at distance ) and fixes a predetermined call order to achieve a -approximation. They prove the approximation bound by case analysis, deriving per-cycle completion times and comparing to lower bounds on the optimal broadcast time, thereby improving the prior best known ratio of 2. The results advance understanding of broadcast time on near-cactus structures and suggest directions for future work, including exact complexity status and potential extensions to broader cactus families or PTAS results.

Abstract

Broadcasting is an information dissemination primitive where a message originates at a node (called the originator) and is passed to all other nodes in the network. Broadcasting research is motivated by efficient network design and determining the broadcast times of standard network topologies. Verifying the broadcast time of a node in an arbitrary network is known to be NP-hard. Additionally, recent findings show that the broadcast time problem is also NP-complete in general cactus graphs and some highly restricted subfamilies of cactus graphs. These graph families are structurally similar to -cycle graphs, in which the broadcast time problem is also believed to be NP-complete. In this paper, we present a simple -approximation algorithm for determining the broadcast time of networks modeled using -cycle graphs, where depends on the structure of the graph.

Paper Structure

This paper contains 7 sections, 4 theorems, 16 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. k-Cycle Graphs
  3. Simple-k-cycle Algorithm
  4. Broadcasting from the Central Vertex
  5. Broadcasting from a Cycle Vertex
  6. Conclusion
  7. Performance of Simple-K-Cycle in Specific Cases

Key Result

lemma thmcounterlemma

(k-cycle-2-approximation). Let $G_k$ be a $k$-cycle graph where the originator is the central vertex $u$. Then

Figures (5)

  • Figure 1: An example of a snowflake graph. A snowflake graph consists of a $k$-cycle graph with paths originating from the vertices adjacent to the central vertex. One vertex on each cycle can also have any number of additional adjacent vertices. Snowflake graphs are cactus graphs with pathwidth at most 2 cactus-telephone-broadcasting.
  • Figure 2: An example of a graph created by the reduction in broadcasting-struct-restrictions. The graph consists of a k-cycle graph with paths connected to the central vertex.
  • Figure 3: An example of a $k$-cycle graph with $k=5$ and $l_1=7, l_2=5, l_3=5, l_4=4, l_5=2$.
  • Figure 4: An example of a broadcast scheme generated by Simple-K-Cycle on a $k$-cycle graph with $l_1=6, l_2=5, l_3=2$ and originator $v_c$. $v_c$ calls cycle $C_1$ in time units 1 and 4, cycle $C_2$ in time units 2 and 5, and cycle $C_3$ in time unit 3. The broadcast is completed in 5 time units.
  • Figure 5: An example of a broadcast scheme generated by Simple-K-Cycle on a $k$-cycle graph with $l_1=9, l_2=8, l_3=4, l_4=2$ and originator $u$ with $dist(u, v_c)=d=2$. $u$ informs towards $v_c$ in time unit 1, then informs its other neighbour in time unit 2. $v_c$ is informed in time unit 2. Then, $v_c$ calls cycle $C_1$ in time units 3 and 6, cycle $C_2$ in time unit 7, cycle $C_3$ in time units 4 and 8, and cycle $C_4$ in time unit 5. The broadcast is completed in 8 time units.

Theorems & Definitions (7)

  • definition thmcounterdefinition
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • proof