用非交换行列式逼近积和式
Approximating the Permanent via Nonabelian Determinants
摘要：
Jerrum、Sinclair和Vigoda的著名工作已经建立了{0，1}矩阵的永久式可以通过使用快速混合马尔可夫链在随机多项式时间内近似。文献的另一部分研究了替代的、纯代数的、多项式时间近似方案的可能性。这些方案的工作原理是将每个1替换为代数A的随机元素，并考虑所得矩阵的行列式。当A是非交换的时，这个行列式可以用几种方式定义。我们证明了，对于基于传统行列式的估计量，当A是d × d矩阵的代数时，二阶矩与一阶矩平方的临界比--因此我们需要获得一个好的估计量的试验次数--是（1 + O（1/d））^n。这些结果可以推广到群代数和一般的半单代数。我们还研究了对称化的行列式的Barvinok，所得到的估计有小的方差时，d是足够大的。然而，对于常数d-唯一已知有效算法的情况-我们证明临界比超过2^{n} / n^{O（d）}。因此，我们的结果并没有提供一个新的多项式时间近似计划的永久。事实上，他们认为，代数方法近似的永久面临重大障碍。我们得到这些结果使用图表技术，我们表示矩阵产品的收缩张量产品。当这些矩阵是随机的，在Haar测度或高斯测度中，我们可以根据适当随机置换的循环结构来计算这些乘积的迹。在对称化的情况下，我们的估计，然后来自一个连接的特征理论的对称群。
英文摘要：
Celebrated work of Jerrum, Sinclair, and Vigoda has established that the permanent of a {0,1} matrix can be approximated in randomized polynomial time by using a rapidly mixing Markov chain. A separate strand of the literature has pursued the possibility of an alternate, purely algebraic, polynomial-time approximation scheme. These schemes work by replacing each 1 with a random element of an algebra A, and considering the determinant of the resulting matrix. When A is noncommutative, this determinant can be defined in several ways. We show that for estimators based on the conventional determinant, the critical ratio of the second moment to the square of the first--and therefore the number of trials we need to obtain a good estimate of the permanent--is (1 + O(1/d))^n when A is the algebra of d by d matrices. These results can be extended to group algebras, and semi-simple algebras in general. We also study the symmetrized determinant of Barvinok, showing that the resulting estimator has small variance when d is large enough. However, for constant d--the only case in which an efficient algorithm is known--we show that the critical ratio exceeds 2^{n} / n^{O(d)}. Thus our results do not provide a new polynomial-time approximation scheme for the permanent. Indeed, they suggest that the algebraic approach to approximating the permanent faces significant obstacles. We obtain these results using diagrammatic techniques in which we express matrix products as contractions of tensor products. When these matrices are random, in either the Haar measure or the Gaussian measure, we can evaluate the trace of these products in terms of the cycle structure of a suitably random permutation. In the symmetrized case, our estimates are then derived by a connection with the character theory of the symmetric group.
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