We provide a primal-dual framework for randomized approximation algorithms utilizing semidefinite programming (SDP) relaxations. Our framework pairs a continuum of APX-complete problems including MaxCut, Max2Sat, MaxDicut, and more generally, Max-Boolean Constraint Satisfaction and MaxQ (maximization of a positive semidefinite quadratic form over the hypercube) with new APX-complete problems which are stated as convex optimization problems with exponentially many variables. These new dual counterparts, based on what we call Grothendieck covers, range from fractional cut covering problems (for MaxCut) to tensor sign covering problems (for MaxQ). For each of these problem pairs, our framework transforms the randomized approximation algorithms with the best known approximation factors for the primal problems to randomized approximation algorithms for their dual counterparts with reciprocal approximation factors which are tight with respect to the Unique Games Conjecture. For each APX-complete pair, our algorithms solve a single SDP relaxation and generate feasible solutions for both problems which also provide approximate optimality certificates for each other. Our work utilizes techniques from areas of randomized approximation algorithms, convex optimization, spectral sparsification, as well as Chernoff-type concentration results for random matrices.

翻译：我们为利用半定规划松弛的随机近似算法提供了一个原始-对偶框架。该框架将一系列APX完全问题（包括MaxCut、Max2Sat、MaxDicut，以及更一般的Max-布尔约束满足问题和MaxQ（在半超立方体上最大化半正定二次型））与表述为具有指数级变量的凸优化问题的新APX完全问题进行配对。这些基于所谓格罗滕迪克覆盖的新对偶问题，涵盖了从分数割覆盖问题（对应MaxCut）到张量符号覆盖问题（对应MaxQ）的范围。对于每一组问题对，我们的框架将针对原始问题具有最知名近似因子的随机近似算法，转化为针对其对偶问题的随机近似算法，其倒数近似因子在唯一游戏猜想下是紧的。对于每一组APX完全问题对，我们的算法求解单个SDP松弛，并为两个问题生成可行解，这些解同时为彼此提供近似最优性证明。我们的工作运用了来自随机近似算法、凸优化、谱稀疏化等领域的技巧，以及针对随机矩阵的切尔诺夫型集中性结果。