Minimizing the Maximum End-to-End Network Delay: Hardness, Algorithm, and Performance

We study a flow problem where a source streams information to a sink over a directed graph G , (V,E) possibly using multiple paths to minimize the maximum end-to-end delay, denoted as Min-Max-Delay problem. Different from the maximum latency problem, our setting is unique in that communication over an edge incurs a constant delay within an associated 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 pseudopolynomial time algorithm for Min-Max-Delay, with complexity of O(log(Ndmax)(Nd max)(logR + Ndmax log(Ndmax))), where N , max{|V |, |E|}, R is the flow rate requirement and dmax is the maximum edge delay. Moreover, we propose a fully polynomial time approximation scheme with a time complexity of O((L+logN) log(M)(NM(logR+NM log(NM))) where M = |E|(1+1/ )+1 and L is the bit complexity for representing dmax. Besides, in this paper we observe that the optimal MinMax-Delay flow can only be achieved by fractional flow solution in some problem instances and in the worst case the Int-Gap, which is defined as the maximum delay gap between the optimal integer Min-Max-Delay flow and the optimal Min-Max-Delay flow, can be infinitely large. We also propose a near-tight bicriteria bound of (1 − , 1/ ) for the Int-Gap. In the end, we show how the results introduced in this paper can be extended to the multi-commodity Min-Max-Delay problem case.