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Online Network Design with Outliers

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Abstract

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 \(\epsilon >0\), select at least \((1-\epsilon )k\) terminals with high probability and pay in expectation \(O(\log ^2n)\) 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 \(\varOmega (\log n/\log \log n)\) lower bound for the case that the online algorithm is allowed to select \(\alpha \,k\) terminals only, for a sufficiently large constant \(\alpha \in (0,1)\).

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Notes

  1. For a weighted graph G, \(dist_G(u,v)\) denotes the distance between nodes u and v in the graph. For the sake of simplicity, we next associate an infinite opening cost to nodes that are not facilities, and let \(\mathcal {F}=V\).

  2. Throughout this paper the expected competitive ratio, also called ratio of expectations (RoE), is the ratio between the expected cost of the solution computed by the online algorithm considered and the expected cost of the optimal offline solution. Sometimes in the literature the expectation of ratios EoR is considered instead (which is typically more complicated).

  3. For the sake of brevity, we will drop the term stochastic from problem names.

  4. Throughout this paper we use the term with high probability (abbreviated whp) to refer to probability that approaches 1 as k, the number of selected terminals, grows. In particular, the probability of failure is polynomially small for \(k=\varOmega (\log n)\) in the considered cases.

  5. The k-MST problem studied in [14] and the Steiner tree problem with outliers are equivalent approximation-wise, modulo constant factors. The same holds for TSP with outliers.

  6. To avoid inessential technicalities, we will always assume that n is a multiple of \(\sigma \).

  7. Since the cost of an MST spanning a set of vertices W is at most twice the corresponding cost of the best Steiner tree connecting those vertices, we can obtain a constant approximation for the outST problem if we have a constant approximation for the rooted k-MST problem. Indeed, by apending a large number N (say \(N=n^2\)) of copies of each terminal with edges of cost 0 and multiplying k by a factor N, we can achieve for outST the same approximation factor as for k-MST, that is 2 [14].

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Acknowledgments

We would like to thank the anonymous reviewers of both this as and the conference version, for their constructive comments.

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Authors and Affiliations

  1. Sapienza University of Rome, Rome, Italy

    Aris Anagnostopoulos, Stefano Leonardi & Piotr Sankowski

  2. DSIA, University of Lugano, Lugano, Switzerland

    Fabrizio Grandoni

  3. University of Warsaw, Warsaw, Poland

    Piotr Sankowski

Authors
  1. Aris Anagnostopoulos
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  2. Fabrizio Grandoni
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  3. Stefano Leonardi
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  4. Piotr Sankowski
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Corresponding author

Correspondence to Aris Anagnostopoulos.

Additional information

A preliminary version of this work was presented in the Proceedings of the 37th International Colloquium on Automata, Languages and Programming (ICALP 2010) [1].

Partially supported by the EU FET projects MULTIPLEX 317532 and SIMPOL 610704, the ERC Starting Grants NEWNET 279352 and PAAl 259515, and the Google Focused Research Award Algorithms for Large-Scale Data Analysis.

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Anagnostopoulos, A., Grandoni, F., Leonardi, S. et al. Online Network Design with Outliers. Algorithmica 76, 88–109 (2016). https://doi.org/10.1007/s00453-015-0021-y

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