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Approximation Algorithms for Cost-robust Discrete Minimization Problems Based on their LP-Relaxations

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Abstract

We consider robust discrete minimization problems where uncertainty is defined by a convex set in the objective. Assuming the existence of an integrality gap verifier with a bounded approximation guarantee for the LP relaxation of the non-robust version of the problem, we derive approximation algorithms for the robust version under different types of uncertainty, including polyhedral and ellipsoidal uncertainty.

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Notes

  1. Yet, there is a third (intermediate) approach, namely, distributionally robust optimization (see, e.g., [17]), in which one optimizes the expectation over the worst-case choice from a set of distributions on the uncertain parameters.

  2. Specifically, we make \(O(\log _{1+\epsilon }n)\) independent runs of the algorithm and return the output with minimum objective.

  3. Using probability amplification techniques, the “o(1)” bound can be made arbitrarily small. Indeed, we can pick \(\ell \) independent samples \({\widehat{x}}^{1},\ldots , {\widehat{x}}^{\ell }\sim {\mathcal {B}}\), return \({\widehat{x}}^i\), where \(i\in {\text {argmin}}\max _{c\in {\mathcal {C}}}c^T{\widehat{x}}^i\), and note that \(\Pr [c^T{\widehat{x}}^{i}\le \alpha \cdot {\textsc {Opt}}_R,\quad \forall c\in {\mathcal {C}}]=1-o(1)^{\ell }\).

  4. Indeed, if \(\Pr _{{\widehat{x}}\sim {\mathcal {B}}}[c^T{\widehat{x}}\le \alpha \cdot {\textsc {Opt}}_R,\quad \forall c\in {\mathcal {C}}]=1-o(1)\), then for any \(c\in {\mathcal {C}}\), \({\mathbb {E}}_{{\widehat{x}}\sim {\mathcal {B}}}[c^T{\widehat{x}}]\le \alpha \cdot {\textsc {Opt}}_R+o(1)\cdot \max _{c\in {\mathcal {C}},x\in {\mathcal {S}}}c^Tx\). Using probability amplification techniques, the error term can be made arbitrarily small.

  5. It would have worked if the function was concave (this is the so-called Jensen’s inequality).

  6. In fact, the notion used in [31] is slightly stronger than the robust-in-expectation notion we use here as the objective there is to find an \(x:={\bar{x}}\) maximizing \(\min _{c\in {\mathcal {C}}}{\mathbb {E}}[c^Tx]\). However since in our analysis (Lemma 1), we compare the obtained solution to \(z^*_R\ge \min _{c\in {\mathcal {C}}}{\mathbb {E}}[c^T{\bar{x}}]\ge {\textsc {Opt}}_R\), we also achieve the same guarantee as in [31].

  7. If \(\beta _j=0\), then it is feasible to set \(u_j=+\infty \), which enforces \(x_j=0\), unless the problem has an unbounded optimum.

  8. Here we use the fact that the constraints in (59) are defined over all \(v\in {\mathbf {E}}(0,D^{-1})\cap {\mathbb {R}}^n_+\), which in turn requires the assumption \(D^{-1}\ge 0\).

  9. Note that \(|T|\le n\).

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Acknowledgements

We thank the anonymous reviewers for the careful reading and useful remarks which helped improve the paper.

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  1. Khalifa University of Science and Technology, SAN Campus, P.O. Box 127788, Abu Dhabi, UAE

    Khaled Elbassioni

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K. Elbassioni: Partially supported by Abu Dhabi Education & Knowledge-Abu Dhabi Award for Research Excellence (AARE18-152)

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Elbassioni, K. Approximation Algorithms for Cost-robust Discrete Minimization Problems Based on their LP-Relaxations. Algorithmica 84, 3622–3654 (2022). https://doi.org/10.1007/s00453-022-00987-z

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