Affine reductions for LPs and SDPs

Abstract

We define a reduction mechanism for LP and SDP formulations that degrades approximation factors in a controlled fashion. Our reduction mechanism is a minor restriction of classical hardness reductions requiring an additional independence assumption and it allows for reusing many hardness reductions that have been used to show inapproximability in the context of PCP theorems. As a consequence we establish strong linear programming inapproximability (for LPs with a polynomial number of constraints) for many problems. In particular we obtain a \(\frac{3}{2}-\varepsilon \) inapproximability for answering an open question in Chan et al. (Proceedings of FOCS, pp. 350–359, 2013, https://doi.org/10.1109/FOCS.2013.45) and prove an inapproximability factor of \(\frac{1}{2}+\varepsilon \) for bounded degree . In the case of SDPs, we obtain inapproximability results for these problems relative to the SDP-inapproximability of \({\textsf {MaxCUT}}_{}\). Moreover, using our reduction framework we are able to reproduce various results for CSPs from Chan et al. (Proceedings of FOCS, pp. 350–359, 2013, https://doi.org/10.1109/FOCS.2013.45) via simple reductions from Max-\(2\)-XOR.

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Acknowledgements

Research reported in this paper was partially supported by NSF Grant CMMI-1300144 and NSF CAREER Grant CMMI-1452463. The authors would like to thank James Lee for the helpful discussions regarding max-CSPs. We are indebted to Siu On Chan for some of the PCP inapproximability bounds as well as Santosh Vempala for the helpful discussions as well as the anonymous reviewers for significantly improving the presentation of the paper.

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Braun, G., Pokutta, S. & Zink, D. Affine reductions for LPs and SDPs. Math. Program. 173, 281–312 (2019). https://doi.org/10.1007/s10107-017-1221-9

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Mathematics Subject Classification

  • 90C22
  • 68Q17
  • 05C70