The limits of SDP relaxations for general-valued CSPs
International audience It has been shown that for a general-valued constraint language Γ the following statements are equivalent: (1) any instance of VCSP(Γ) can be solved to optimality using a constant level of the Sherali-Adams LP hierarchy; (2) any instance of VCSP(Γ) can be solved to optimality...
Published in: | 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) |
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Main Authors: | , |
Other Authors: | , , |
Format: | Conference Object |
Language: | English |
Published: |
HAL CCSD
2017
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Subjects: | |
Online Access: | https://hal-upec-upem.archives-ouvertes.fr/hal-01796715 https://doi.org/10.1109/LICS.2017.8005087 |
Summary: | International audience It has been shown that for a general-valued constraint language Γ the following statements are equivalent: (1) any instance of VCSP(Γ) can be solved to optimality using a constant level of the Sherali-Adams LP hierarchy; (2) any instance of VCSP(Γ) can be solved to optimality using the third level of the Sherali-Adams LP hierarchy; (3) the support of Γ satisfies the " bounded width condition " , i.e., it contains weak near-unanimity operations of all arities. We show that if the support of Γ violates the bounded with condition then not only is VCSP(Γ) not solved by a constant level of the Sherali-Adams LP hierarchy but it is also not solved by Ω(n) levels of the Lasserre SDP hierarchy (also known as the sum-of-squares SDP hierarchy). For Γ corresponding to linear equations in an Abelian group, this result follows from existing work on inapproximability of Max-CSPs. By a breakthrough result of Lee, Raghavendra, and Steurer [STOC'15], our result implies that for any Γ whose support violates the bounded width condition no SDP relaxation of polynomial-size solves VCSP(Γ). We establish our result by proving that various reductions preserve exact solvability by the Lasserre SDP hierarchy (up to a constant factor in the level of the hierarchy). Our results hold for general-valued constraint languages, i.e., sets of functions on a fixed finite domain that take on rational or infinite values, and thus also hold in notable special cases of {0, ∞}-valued languages (CSPs), {0, 1}-valued languages (Min-CSPs/Max-CSPs), and Q-valued languages (finite-valued CSPs). |
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