关于有界路宽/路宽矩阵的积和式与完美匹配的表现力
On the expressive power of permanents and perfect matchings of matrices of bounded pathwidth/cliquewidth
摘要：
大约25年前，Valiant引入了一种代数计算模型，以研究多项式族的复杂性。该理论是与复杂性类VP和VNP一起引入的，它们是经典类P和NP的类似物。多项式族难以计算（即VNP-完全），包括永久多项式和哈密顿多项式。在以前的论文中，作者与P. Koiran一起研究了有界树宽矩阵的永久多项式和Hamilton多项式的表达能力，以及平面图的完美匹配的表达能力。证明了有界树宽矩阵的积和多项式与算术公式的等价性。同时，证明了平面图的完美匹配的权和等价于（弱）扭回路。在本文中，我们继续上述方向的研究，并研究的表现能力的永久式，汉密尔顿和完美匹配的矩阵有界的路宽或有界的宽度。特别地，我们证明了具有有界路径宽度的矩阵的积和式、哈密顿量和完美匹配精确地表示算术公式。这是我们以前的结果有界树宽矩阵的一个改进。此外，矩阵的有界加权的cnc宽度，我们显示这些多项式的成员VP。
英文摘要：
Some 25 years ago Valiant introduced an algebraic model of computation in order to study the complexity of evaluating families of polynomials. The theory was introduced along with the complexity classes VP and VNP which are analogues of the classical classes P and NP. Families of polynomials that are difficult to evaluate (that is, VNP-complete) includes the permanent and hamiltonian polynomials. In a previous paper the authors together with P. Koiran studied the expressive power of permanent and hamiltonian polynomials of matrices of bounded treewidth, as well as the expressive power of perfect matchings of planar graphs. It was established that the permanent and hamiltonian polynomials of matrices of bounded treewidth are equivalent to arithmetic formulas. Also, the sum of weights of perfect matchings of planar graphs was shown to be equivalent to (weakly) skew circuits. In this paper we continue the research in the direction described above, and study the expressive power of permanents, hamiltonians and perfect matchings of matrices that have bounded pathwidth or bounded cliquewidth. In particular, we prove that permanents, hamiltonians and perfect matchings of matrices that have bounded pathwidth express exactly arithmetic formulas. This is an improvement of our previous result for matrices of bounded treewidth. Also, for matrices of bounded weighted cliquewidth we show membership in VP for these polynomials.
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