Notations

Let G = (V, E) be an undirected graph with nonnegative capacities c(e) on edges e ∈ E. Suppose there are k source-destination pairs (s _i , t _i ) for i = 1, …, k, and let d _i denote the amount of flow (or demand) that pair i wishes to send from s _i to t _i . Given a routing of these flows on G, the congestion of an edge e is defined as u(e)/c(e), the ratio of the total flow crossing edge e divided by its capacity. The congestion of the overall routing is defined as the maximum congestion over all edges. The congestion minimization problem is to find the routing that minimizes the maximum congestion. Observe that specifying a flow from s _i to t _i is equivalent to finding a probability distribution (not necessarily unique) on a collection of paths from s _i to t _i .

The congestion minimization problem can be studied in many settings. In the offline setting, the instance of the flow problem is provided in advance, and the goal is to find the optimum routing. In the online setting, the demands arrive in an arbitrary adversarial order, and a flow must be specified for a demand immediately upon arrival; this flow is fixed forever and cannot be rerouted later when new demands arrive. Several distributed approaches have also been studied where each pair routes its flow in a distributed manner based on some global information such as the current congestion on the edges.

In this note, the oblivious setting is considered. Here, a routing scheme is specified for each pair of vertices in advance without any knowledge of which demands will actually arrive. Note that an algorithm in the oblivious setting is severely restricted. In particular, if d _i units of demand arrive for pair (s _i ,t _i ), the algorithm must necessarily route this demand according to the prespecified paths irrespective of the other demands or any other information such as congestion of other edges. Thus, given a network graph G, the oblivious flows need to be computed just once. After this is done, the job of the routing algorithm is trivial; whenever a demand arrives, it simply routes it along the precomputed path. An oblivious routing scheme is called ccompetitive if for any collection of demands D, the maximum congestion of the oblivious routing is no more than c times the congestion of the optimum offline solution for D. Given this stringent requirement on the quality of oblivious routing, it is not a priori clear that any reasonable oblivious routing scheme should exist at all.

Techniques

This section describes the high-level idea of Räcke’s result. For a subset S ⊂ V , let cap(S) denote the total capacity of the edges that cross the cut (S,V ∖S), and let dem(S) denote the total demand that must be routed across the cut (S,V ∖S). Observe that q = max _S ⊂ Vdem(S)/cap(S) is a lower bound on the congestion of any solution. On the other hand, the key result [3, 13] relating multicommodity flows and cuts implies that there is a routing such that the maximum congestion is at most O(qlog k) where k is the number of distinct source sink pairs. However, note that this by itself does not suffice to obtain good oblivious routings, since a pair (s _i ,t _i ) can have different routing for different demand sets. The main idea of Räcke was to impose a treelike structure for routing on the graph to achieve obliviousness. This is formalized by a hierarchical decomposition described below.

Consider a hierarchical decomposition of the graph G = (V,E) as follows.Starting from the set S = V, the sets are partitioned successively until each set becomes singleton vertex. This hierarchical decomposition can be viewed naturally as a tree T, where the root corresponds to the set V and leaves corresponds to the singleton sets {v}. Let S _i denote the subset of V corresponding to node i in T. For an edge (i,j) in the tree where i is the child of j, assign it a capacity equal to cap(S _i ) (note that this is the capacity from S _i to the rest of G and not just capacity between S _i and S _j in G). The tree T is used to simulate routing in G and vice versa. Given a demand from u to v in G, consider the corresponding (unique) route among leaves corresponding to {u} and {v} in T. For any set of demands, it is easily seen that the congestion in T is no more than the congestion in G. Conversely, Räcke showed that there also exists a tree T where the routes in T can be mapped back to flows in G, such that for any set of demands, the congestion in G is at most O(log ³n) times that in T. In this mapping, a flow along the (i,j) in the tree T corresponds to a suitably constructed flow between sets S _i and S _j in G. Since route between any two vertices in T is unique, this gives an oblivious routing in G.

Räcke uses very clever ideas to show the existence of such a hierarchical decomposition. Describing the construction is beyond the scope of this note, but it is instructive to understand the properties that must be satisfied by such a decomposition. First, the tree T should capture the bottlenecks in G, i.e., if there is a set of demands that produces high congestion in G, then it should also produce a high congestion in T. A natural approach to construct T would be to start with V, split V along a bottleneck (formally, along a cut with low sparsity), and recurse. However, this approach is too simple to work. As discussed below, T must also satisfy two other natural conditions, known as the bandwidth property and the weight property which are motivated as follows. Consider a node i connected to its parent j in T. Then, i needs to route dem(S _i ) flow out of S _i , and it incurs congestion dem(S _i )/cap(S _i ) in T. However, when T is mapped back to G, all the flow going out of S _i must pass via S _j . To ensure that the edges from S _i to S _j are not overloaded, it must be the case that the capacity from S _i to S _j is not too small compared to the capacity from S _i to the rest of the graph V ∖S _i . This is referred to as the bandwidth property. Räcke guarantees that this is ratio is always Ω(1∕logn) for every S _i and S _j corresponding to edges (i, j) in the tree. The weight property is motivated as follows. Consider a node j in T with children i₁, …, i _p ; then the weight property essentially requires that the sets S_i1, …, S _ip should be well connected among themselves even when restricted to the subgraph S _j . To see why this is needed, consider any communication between, say, nodes i₁ and i₂ in T. It takes the route i₁ to j to i₂, and hence, in G,S_i1 cannot use edges that lie outside S _j to communicate with S_i2. Räcke shows that these conditions suffice and that a decomposition can be obtained that satisfies them.

The factor O(log ³n) in Räcke’s guarantee arises from three sources. The first logarithmic factor is due to the flow-cut gap [3, 13]. The second is due to the logarithmic height of the tree, and the third is due to the loss of a logarithmic factor in the bandwidth and weight properties.