We study integrality gap (IG) lower bounds on strong LP and SDP relaxations derived by the Sherali-Adams (SA), Lovasz-Schrijver-SDP (LS+), and Sherali-Adams-SDP (SA+) lift-and-project (L&P) systems for the t-Partial-Vertex-Cover (t-PVC) problem, a variation of the classic Vertex-Cover problem in which only t edges need to be covered. t-PVC admits a 2-approximation using various algorithmic techniques, all relying on a natural LP relaxation. Starting from this LP relaxation, our main results assert that for every epsilon > 0, level-Theta(n) LPs or SDPs derived by all known L&P systems that have been used for positive algorithmic results (but the Lasserre hierarchy) have IGs at least (1-epsilon)n/t, where n is the number of vertices of the input graph. Our lower bounds are nearly tight. Our results show that restricted yet powerful models of computation derived by many L&P systems fail to witness c-approximate solutions to t-PVC for any constant c, and for t = O(n). This is one of the very few known examples of an intractable combinatorial optimization problem for which LP-based algorithms induce a constant approximation ratio, still lift-and-project LP and SDP tightenings of the same LP have unbounded IGs. We also show that the SDP that has given the best algorithm known for t-PVC has integrality gap n/t on instances that can be solved by the level-1 LP relaxation derived by the LS system. This constitutes another rare phenomenon where (even in specific instances) a static LP outperforms an SDP that has been used for the best approximation guarantee for the problem at hand. Finally, one of our main contributions is that we make explicit of a new and simple methodology of constructing solutions to LP relaxations that almost trivially satisfy constraints derived by all SDP L&P systems known to be useful for algorithmic positive results (except the La system).

中文翻译：

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在部分顶点覆盖多面体上执行的提升和项目系统

我们研究了由 Sherali-Adams (SA)、Lovasz-Schrijver-SDP (LS+) 和 Sherali-Adams-SDP (SA+)lift-and-project (L&P) 导出的强 LP 和 SDP 松弛的完整性间隙 (IG) 下界) t-Partial-Vertex-Cover (t-PVC) 问题的系统，经典的 Vertex-Cover 问题的变体，其中只需要覆盖 t 条边。t-PVC 允许使用各种算法技术进行 2-近似，所有这些都依赖于自然的 LP 松弛。从这个 LP 松弛开始，我们的主要结果断言，对于每个 epsilon > 0，由所有已知的 L&P 系统导出的水平 Theta(n) LP 或 SDP 已用于正算法结果（但 Lasserre 层次结构）至少具有 IG (1-epsilon)n/t，其中 n 是输入图的顶点数。我们的下限几乎很紧。我们的结果表明，由许多 L&P 系统导出的受限但强大的计算模型无法证明任何常数 c 和 t = O(n) 的 t-PVC 的 c 近似解。这是难以处理的组合优化问题的极少数已知示例之一，基于 LP 的算法会为其引入恒定的近似比，但相同 LP 的提升和投影 LP 和 SDP 收紧具有无限的 IG。我们还表明，给出 t-PVC 已知最佳算法的 SDP 在实例上具有完整性间隙 n/t，可以通过 LS 系统导出的 1 级 LP 松弛来解决。这构成了另一种罕见的现象，其中（即使在特定情况下）静态 LP 优于已用于手头问题的最佳近似保证的 SDP。最后，