The Inapproximability of Maximum Single-Sink Unsplittable, Priority and Confluent Flow Problems [article]

F. Bruce Shepherd, Adrian Vetta
2015 arXiv   pre-print
We consider single-sink network flow problems. An instance consists of a capacitated graph (directed or undirected), a sink node t and a set of demands that we want to send to the sink. Here demand i is located at a node s_i and requests an amount d_i of flow capacity in order to route successfully. Two standard objectives are to maximise (i) the number of demands (cardinality) and (ii) the total demand (throughput) that can be routed subject to the capacity constraints. Furthermore, we examine
more » ... these maximisation problems for three specialised types of network flow: unsplittable, confluent and priority flows. In the unsplittable flow problem (UFP), we have edge capacities, and the demand for s_i must be routed on a single path. In the confluent flow problem, we have node capacities, and the final flow must induce a tree. Both of these problems have been studied extensively, primarily in the single-sink setting. However, most of this work imposed the no-bottleneck assumption (that the maximum demand d_max is at most the minimum capacity u_min). Given the no-bottleneck assumption (NBA), there is a factor 4.43-approximation algorithm due to Dinitz et al. for the unsplittable flow problem. Under the stronger assumption of uniform capacities, there is a factor 3-approximation algorithm due to Chen et al. for the confluent flow problem. However, unlike the UFP, we show that a constant factor approximation algorithm cannot be obtained for the single-sink confluent flows even with the NBA. Without NBA, we show that maximum cardinality single-sink UFP is hard to approximate to within a factor n^.5-ϵ even when all demands lie in a small interval [1,1+Δ] where Δ>0 (but has polynomial input size). This is very sharp since when Δ=0, this becomes a maximum flow problem.
arXiv:1504.00627v2 fatcat:i2fmgalrgfgh7hfsry4rnugk74