Online Network Design with Outliers

In a classical online network design problem, traffic requirements are gradually revealed to an algorithm. Each time a new request arrives, the algorithm has to satisfy it by augmenting the network under construction in a proper way (with no possibility of recovery). In this paper we study a natural generalization of online network design problems, where a fraction of the requests (the outliers) can be disregarded. Now, each time a request arrives, the algorithm first decides whether to satisfy it or not, and only in the first case it acts accordingly. We cast three classical network design problems into this framework: (i) Online Steiner tree with outliers In this case a set of t terminals that belong to an n-node graph is presented, one at a time, to an algorithm. Each time a new terminal arrives, the algorithm can either discard or select it. In the latter case, the algorithm connects it to the Steiner tree under construction (initially consisting of a given root node). At the end of the process, at least k terminals must be selected. (ii) Online TSP with outliers This is the same problem as above, but with the Steiner tree replaced by a TSP tour. (iii) Online facility location with outliers In this case, we are also given a set of facility nodes, each one with an opening cost. Each time a terminal is selected, we have to connect it to some facility (and open that facility, if it is not already open). We focus on the known distribution model, where terminals are independently sampled from a given distribution. For all the above problems, we present bicriteria online algorithms that, for any constant $$\epsilon >0$$ϵ>0, select at least $$(1-\epsilon )k$$(1-ϵ)k terminals with high probability and pay in expectation $$O(\log ^2n)$$O(log2n) times more than the expected cost of the optimal offline solution (selecting k terminals). These upper bounds are complemented by inapproximability results for the case that one insists on selecting exactly k terminals, and by lower bounds including an $$\varOmega (\log n/\log \log n)$$Ω(logn/loglogn) lower bound for the case that the online algorithm is allowed to select $$\alpha \,k$$αk terminals only, for a sufficiently large constant $$\alpha \in (0,1)$$α∈(0,1).