Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: It has been shown that for a general-valued constraint language $\Gamma$ the following statements are equivalent: (1) any instance of $\operatorname{VCSP}(\Gamma)$ can be solved to optimality using a constant level of the Sherali-Adams LP hierarchy; (2) any instance of $\operatorname{VCSP}(\Gamma)$ can be solved to optimality using the third level of the Sherali-Adams LP hierarchy; (3) the support of $\Gamma$ satisfies the "bounded width condition", i.e., it contains ak near-unanimity operations of all arities. #R##N# show that if the support of $\Gamma$ violates the bounded with condition then not only is $\operatorname{VCSP}(\Gamma)$ not solved by a constant level of the Sherali-Adams LP hierarchy but it is also not solved by $\Omega(n)$ levels of the Lasserre SDP hierarchy (also known as the sum-of-squares SDP hierarchy). For $\Gamma$ 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 $\Gamma$ whose support violates the bounded width condition no SDP relaxation of polynomial-size solves $\operatorname{VCSP}(\Gamma)$. #R##N# 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 $\{0,\infty\}$-valued languages (CSPs), $\{0,1\}$-valued languages (Min-CSPs/Max-CSPs), and $\mathbb{Q}$-valued languages (finite-valued CSPs).
['Johan Thapper', 'Stanislav Zivny']
I am writing a research paper on a idea The limits of SDP relaxations for general-valued CSPs
[{"from": "human", "value": "I am writing a research paper on a idea The limits of SDP relaxations for general-valued CSPs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: It has been shown that for a general-valued constraint language $\\Gamma$ the following statements are equivalent: (1) any instance of $\\operatorname{VCSP}(\\Gamma)$ can be solved to optimality using a constant level of the Sherali-Adams LP hierarchy; (2) any instance of $\\operatorname{VCSP}(\\Gamma)$ can be solved to optimality using the third level of the Sherali-Adams LP hierarchy; (3) the support of $\\Gamma$ satisfies the \"bounded width condition\", i.e., it contains ak near-unanimity operations of all arities. #R##N# show that if the support of $\\Gamma$ violates the bounded with condition then not only is $\\operatorname{VCSP}(\\Gamma)$ not solved by a constant level of the Sherali-Adams LP hierarchy but it is also not solved by $\\Omega(n)$ levels of the Lasserre SDP hierarchy (also known as the sum-of-squares SDP hierarchy). For $\\Gamma$ 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 $\\Gamma$ whose support violates the bounded width condition no SDP relaxation of polynomial-size solves $\\operatorname{VCSP}(\\Gamma)$. #R##N# 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 $\\{0,\\infty\\}$-valued languages (CSPs), $\\{0,1\\}$-valued languages (Min-CSPs/Max-CSPs), and $\\mathbb{Q}$-valued languages (finite-valued CSPs)."}]