Oblivious network design

Abstract

Consider the following network design problem: given a network G = (V, E), source-sink pairs {si, ti} arrive and desire to send a unit of flow between themselves. The cost of the routing is this: if edge e carries a total of fe flow (from all the terminal pairs), the cost is given by Σ el(fe), where l is some concave cost function; the goal is to minimize the total cost incurred. However, we want the routing to be oblivious: when terminal pair {si, ti} makes its routing decisions, it does not know the current flow on the edges of the network, nor the identity of the other pairs in the system. Moreover, it does not even know the identity of the l, merely knowing that l is a concave of the total flow on the edge. How should it (obliviously) route its one unit of flow? Can we get competitive algorithms for this problem?In this paper, we develop a framework to model oblivious network design problems (of which the above problem is a special case), and give algorithms with poly-logarithmic competitive ratio for problems in this framework (and hence for this problem). Abstractly, given a problem like the one above, the solution is a multicommodity flow producing a on each edge of Le = l(f1(e),f2(e), ..., fk(e)), and the total cost is given by an function agg (Le1,...,Lem) of the loads of all edges. Our goal is to develop oblivious algorithms that approximately minimize the total cost of the routing, knowing the aggregation agg, but merely knowing that l lies in some class C, and having no other information about the current state of the network. Hence we want algorithms that are simultaneously function-oblivious as well as traffic-oblivious.The aggregation functions we consider are the max and σ objective functions, which correspond to the well-known measures of congestion and total cost of a network; in this paper, we prove the following:• If the aggregation is Σ, we give an oblivious algorithm with O(log2n) competitive ratio whenever the load l is in the class of monotone sub-additive functions. (Recall that our algorithm is also function-oblivious; it works whenever each edge has a load l in the class.)• For the case when the aggregation is max, we give an oblivious algorithm with O(log2n log log n) competitive ratio, when the load l is a norm; we also show that such a competitive ratio is not possible for general sub-additive functions.These are the first such general results about oblivious algorithms for network design problems, and we hope the ideas and techniques will lead to more and improved results in this area.