关于矩阵积式的符号计算方法
On Symbolic Approaches for Computing the Matrix Permanent
摘要：
计算二部图中完美匹配的个数，或者等价地计算0-1矩阵的积和式，是一个重要的组合问题，已经被理论家和实践者广泛研究。永久式是#P-Complete;因此不太可能存在多项式时间算法来解决这个问题。因此，研究人员专注于寻找易于处理的矩阵子类进行永久计算。其中一个备受关注的子类是稀疏矩阵，即只有少数元素被设置为1，其余元素为0的矩阵。对于这个子类，改进的理论上界和实际有效的算法已经开发出来。在本文中，我们问是否有可能超越稀疏矩阵，在我们寻求发展可扩展的技术，为永久，并肯定地回答这个问题。我们的关键见解是用代数决策图（ADD）象征性地表示永久计算。基于ADD的技术自然地使用动态规划，并且因此通过记忆避免冗余计算。这允许利用隐藏的结构在一个大类的矩阵，迄今为止仍然超出了永久计算技术的范围。实现ADD的复杂库的可用性也使得工程实用解决方案的任务相对简单。虽然一个完整的表征矩阵承认一个紧凑的ADD表示仍然是开放的，我们提供了强有力的实验证据，我们的方法的有效性计算的永久性，不仅为稀疏矩阵，而且为密集矩阵和矩阵的“类似”行。
英文摘要：
Counting the number of perfect matchings in bipartite graphs, or equivalently computing the permanent of 0-1 matrices, is an important combinatorial problem that has been extensively studied by theoreticians and practitioners alike. The permanent is #P-Complete; hence it is unlikely that a polynomial-time algorithm exists for the problem. Researchers have therefore focused on finding tractable subclasses of matrices for permanent computation. One such subclass that has received much attention is that of sparse matrices i.e. matrices with few entries set to 1, the rest being 0. For this subclass, improved theoretical upper bounds and practically efficient algorithms have been developed. In this paper, we ask whether it is possible to go beyond sparse matrices in our quest for developing scalable techniques for the permanent, and answer this question affirmatively. Our key insight is to represent permanent computation symbolically using Algebraic Decision Diagrams (ADDs). ADD-based techniques naturally use dynamic programming, and hence avoid redundant computation through memoization. This permits exploiting the hidden structure in a large class of matrices that have so far remained beyond the reach of permanent computation techniques. The availability of sophisticated libraries implementing ADDs also makes the task of engineering practical solutions relatively straightforward. While a complete characterization of matrices admitting a compact ADD representation remains open, we provide strong experimental evidence of the effectiveness of our approach for computing the permanent, not just for sparse matrices, but also for dense matrices and for matrices with "similar" rows.
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