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

Abstract

We consider the single-sink network flow problem. 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 these maximisation problems for three specialised types of network flow: unsplittable, confluent and priority flows.In the unsplittable flow problem, 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, there is a factor 4.43-approximation algorithm due to Dinitz et al. [16] for the unsplittable flow problem. Under the even stronger assumption of uniform capacities, there is a factor 3-approximation algorithm due to Chen et al. [10] for the confluent flow problem. However, unlike the unsplittable flow problem, a constant factor approximation algorithm cannot be obtained for the single-sink confluent flow problem even with the no-bottleneck assumption. Specifically, we prove that it is hard in that setting to approximate single-sink confluent flow to within O(log 1− (n)), for any > 0. This result applies for both cardinality and throughput objectives even in undirected graphs.The remainder of our results focus upon the setting without the no-bottleneck assumption. There, the only result we are aware of is an Ω(m 1− ) inapproximability result of Azar and Regev [3] for cardinality single-sink unsplittable flow in directed graphs. We prove this lower bound applies to undirected graphs, including planar networks. This is the first super-constant hardness known for undirected single-sink unsplittable flow, and apparently the first polynomial hardness for undirected unsplittable flow even for general

Minmax Regret for sink location on paths with general capacities

“…Min-cost confluent flows are hard to construct in both the static and dynamic cases [13,15,23]; Even finding a constant factor approximate solution in the 1-sink case is NP-Hard.…”