Tutte多项式的不可逼近性
Inapproximability of the Tutte polynomial
摘要：
图G的Tutte多项式是一个二元多项式T（G;x，y），它编码了图的许多有趣的性质。我们研究了以下问题的复杂性，对于有理数x和y：将图G作为输入，并输出一个值，该值是T（G;x，y）的一个很好的近似。Jaeger，Vertigan和Welsh已经完全映射了精确计算Tutte多项式的复杂性。他们证明了这是P-难的，除了沿着双曲线（x-1）（y-1）=1和在四个特殊点。我们感兴趣的是确定哪些点（x，y）存在T（G;x，y）的“完全多项式随机逼近方案”（FPRAS）。在RP不等于NP的假设下，我们证明了如果（x，y）在半平面x<-1或y<-1中，则在（x，y）处不存在FPRAS（不包括上面提到的容易计算的情况）。这个结果的两个例外是半直线x<-1，y=1（它仍然是开放的）和双曲线（x-1）（y-1）=2对应于y<-1的部分，我们证明了它们在困难程度上等价于近似计数完美匹配。我们进一步的棘手的结果（x，y）在附近的原点。我们的结果的一个推论是，在RP不等于NP的假设下，当λ>2是正整数时，在点（x，y）=（0，1-λ）处不存在FPRAS.因此，对于λ>2，没有FPRAS来计数非零λ流。这是我们工作的一个有趣的结果，因为相应的决策问题是在P中，例如λ=6。
英文摘要：
The Tutte polynomial of a graph G is a two-variable polynomial T(G;x,y) that encodes many interesting properties of the graph. We study the complexity of the following problem, for rationals x and y: take as input a graph G, and output a value which is a good approximation to T(G;x,y). Jaeger, Vertigan and Welsh have completely mapped the complexity of exactly computing the Tutte polynomial. They have shown that this is #P-hard, except along the hyperbola (x-1)(y-1)=1 and at four special points. We are interested in determining for which points (x,y) there is a "fully polynomial randomised approximation scheme" (FPRAS) for T(G;x,y). Under the assumption RP is not equal to NP, we prove that there is no FPRAS at (x,y) if (x,y) is in one of the half-planes x<-1 or y<-1 (excluding the easy-to-compute cases mentioned above). Two exceptions to this result are the half-line x<-1, y=1 (which is still open) and the portion of the hyperbola (x-1)(y-1)=2 corresponding to y<-1 which we show to be equivalent in difficulty to approximately counting perfect matchings. We give further intractability results for (x,y) in the vicinity of the origin. A corollary of our results is that, under the assumption RP is not equal to NP, there is no FPRAS at the point (x,y)=(0,1--lambda) when λ>2 is a positive integer. Thus there is no FPRAS for counting nowhere-zero λflows for λ>2. This is an interesting consequence of our work since the corresponding decision problem is in P for example for λ=6.
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