我们展示了一种新的方法，轮向量解决方案的半定规划（SDP）层次到积分的解决方案，这些层次之间的连接和输入图的频谱的基础上。我们证明了我们的方法的实用性，提供了一个新的SDP层次为基础的算法的约束满足问题的2变量约束（2-CSP的）。更具体地说，我们为每个2-CSP实例I示出了对于I的Lasserre SDP层次的r轮的舍入算法，其获得至多比松弛值差的\eps的积分解（归一化为位于[0，1]中），只要r > k \cdot \rank_{\geq θ}（\Ins）/ \poly（\e）\;其中k是I，$θ=\poly(\e/k)$的字母大小，并且$\rank_{\geq θ}(\Ins)$表示$\Ins$的约束图的归一化邻接矩阵中大于$θ$的特征值的数目。在$\Ins$是\uniquegames实例的情况下，阈值$θ$仅是$\e$中的多项式，并且与字母表大小无关。同样在这种情况下，我们可以给出\emph{every}实例的轮数的非平凡界。特别是，我们的结果产生了一个基于SDP层次的算法，在最坏的情况下，该算法与Arora，Barak和Steurer（FOCS 2010）最近的次指数算法的性能相匹配，但在自然的实例家族上运行得更快，从而进一步限制了Khot唯一游戏猜想的可能硬实例集。我们的算法实际上需要的$n^{O(r)}$的Lasserre层次结构的$r^{th}$级别指定的约束，并在某些情况下，$r$轮我们的程序可以在时间$2^{O(r)}\poly(n)$评估。
英文摘要：
We show a new way to round vector solutions of semidefinite programming (SDP) hierarchies into integral solutions, based on a connection between these hierarchies and the spectrum of the input graph. We demonstrate the utility of our method by providing a new SDP-hierarchy based algorithm for constraint satisfaction problems with 2-variable constraints (2-CSP's).
More concretely, we show for every 2-CSP instance I a rounding algorithm for r rounds of the Lasserre SDP hierarchy for I that obtains an integral solution that is at most \eps worse than the relaxation's value (normalized to lie in [0,1]), as long as r > k\cdot\rank_{\geq θ}(\Ins)/\poly(\e) \;, where k is the alphabet size of I, $θ=\poly(\e/k)$, and $\rank_{\geq θ}(\Ins)$ denotes the number of eigenvalues larger than $θ$ in the normalized adjacency matrix of the constraint graph of $\Ins$.
In the case that $\Ins$ is a \uniquegames instance, the threshold $θ$ is only a polynomial in $\e$, and is independent of the alphabet size. Also in this case, we can give a non-trivial bound on the number of rounds for \emph{every} instance. In particular our result yields an SDP-hierarchy based algorithm that matches the performance of the recent subexponential algorithm of Arora, Barak and Steurer (FOCS 2010) in the worst case, but runs faster on a natural family of instances, thus further restricting the set of possible hard instances for Khot's Unique Games Conjecture.
Our algorithm actually requires less than the $n^{O(r)}$ constraints specified by the $r^{th}$ level of the Lasserre hierarchy, and in some cases $r$ rounds of our program can be evaluated in time $2^{O(r)}\poly(n)$.
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We give a quantum speedup for solving the canonical semidefinite programming relaxation for binary quadratic optimization. This class of relaxations for combinatorial optimization has so far eluded quantum speedups. Our methods combine ideas from quantum Gibbs sampling and matrix exponent updates. A de-quantization of the algorithm also leads to a faster classical solver. For generic instances, our quantum solver gives a nearly quadratic speedup over state-of-the-art algorithms. Such instances include approximating the ground state of spin glasses and MaxCut on Erdös-Rényi graphs. We also prov…
