Non-approximability and Polylogarithmic Approximations of the Single-Sink Unsplittable and Confluent Dynamic Flow Problems

Dynamic Flows were introduced by Ford and Fulkerson in 1958 to model flows over time. They define edge capacities to be the total amount of flow that can enter an edge {\em in one time unit}. Each edge also has a length, representing the time needed to traverse it. Dynamic Flows have been used to model many problems including traffic congestion, hop-routing of packets and evacuation protocols in buildings. While the basic problem of moving the maximal amount of supplies from sources to sinks is polynomial time solvable, natural minor modifications can make it NP-hard. One such modification is that flows be confluent, i.e., all flows leaving a vertex must leave along the same edge. This corresponds to natural conditions in, e.g., evacuation planning and hop routing. We investigate the single-sink Confluent Quickest Flow problem. The input is a graph with edge capacities and lengths, sources with supplies and a sink. The problem is to find a confluent flow minimizing the time required to send supplies to the sink. Our main results include: a) Logarithmic Non-Approximability: Directed Confluent Quickest Flows cannot be approximated in polynomial time with an $O(\log n)$ approximation factor, unless $P=NP$. b) Polylogarithmic Bicriteria Approximations: Polynomial time $(O(\log^8 n), O(\log^2 κ))$ bicritera approximation algorithms for the Confluent Quickest Flow problem where $κ$ is the number of sinks, in both directed and undirected graphs. Corresponding results are also developed for the Confluent Maximum Flow over timeproblem. The techniques developed also improve recent approximation algorithms for static confluent flows.