Optimizing Distances for Multi-Broadcast in Temporal Graphs

TL;DR

This paper introduces the D-Temporal Multi-Broadcast problem in temporal graphs, analyzing its complexity, approximability, and providing algorithms for various distance measures, with implications for network scheduling and information dissemination.

Contribution

It formalizes the D-TMB problem, characterizes its computational complexity under different distance metrics, and offers approximation algorithms and structural conditions for tractability.

Findings

D-TMB generalizes ReachFast problem.

Polynomial solutions for EA and LD distances with a single source.

NP-hardness and inapproximability results for other distance measures.

Abstract

Temporal graphs represent networks in which connections change over time, with edges available only at specific moments. Motivated by applications in logistics, multi-agent information spreading, and wireless networks, we introduce the D-Temporal Multi-Broadcast (D-TMB) problem, which asks for scheduling the availability of edges so that a predetermined subset of sources reach all other vertices while optimizing the worst-case temporal distance D from any source. We show that D-TMB generalizes ReachFast (arXiv:2112.08797). We then characterize the computational complexity and approximability of D-TMB under six definitions of temporal distance D, namely Earliest-Arrival (EA), Latest-Departure (LD), Fastest-Time (FT), Shortest-Traveling (ST), Minimum-Hop (MH), and Minimum-Waiting (MW). For a single source, we show that D-TMB can be solved in polynomial time for EA and LD, while for the other temporal distances it is NP-hard and hard to approximate within a factor that depends on the adopted distance function. We give approximation algorithms for FT and MW. For multiple sources, if feasibility is not assumed a priori, the problem is inapproximable within any factor unless P = NP, even with just two sources. We complement this negative result by identifying structural conditions that guarantee tractability for EA and LD for any number of sources.