On the Min-Max-Delay Problem: NP-completeness, Algorithm, and Integrality Gap

TL;DR

This paper investigates the computational complexity of the Min-Max-Delay problem in delay-sensitive networks, proving NP-completeness, providing an optimal algorithm, and analyzing the integrality gap between fractional and integer solutions.

Contribution

It establishes NP-completeness results, introduces an optimal pseudo-polynomial algorithm, and analyzes the integrality gap for the Min-Max-Delay problem.

Findings

Min-Max-Delay is weakly NP-complete.

It becomes strongly NP-complete with integer flow constraints.

An optimal pseudo-polynomial time algorithm is proposed.

Abstract

We study a delay-sensitive information flow problem where a source streams information to a sink over a directed graph G(V,E) at a fixed rate R possibly using multiple paths to minimize the maximum end-to-end delay, denoted as the Min-Max-Delay problem. Transmission over an edge incurs a constant delay within the capacity. We prove that Min-Max-Delay is weakly NP-complete, and demonstrate that it becomes strongly NP-complete if we require integer flow solution. We propose an optimal pseudo-polynomial time algorithm for Min-Max-Delay, with time complexity O(\log (Nd_{\max}) (N^5d_{\max}^{2.5})(\log R+N^2d_{\max}\log(N^2d_{\max}))), where N = \max\{|V|,|E|\} and d_{\max} is the maximum edge delay. Besides, we show that the integrality gap, which is defined as the ratio of the maximum delay of an optimal integer flow to the maximum delay of an optimal fractional flow, could be arbitrarily large.