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平面图的Tutte多项式的不可逼近性

Inapproximability of the Tutte polynomial of a planar graph

Leslie Ann Goldberg, Mark Jerrum

备注：
In this revision, significant changes have been made to the structure of the paper to clarify the presentation. Extra detail has been provided at certain points in the proofs, and on at least one occasion the exposition has been made more systematic. Some typos have been corrected. The results are unchanged. 28 pages and 5 figures
发表信息：
Computational Complexity, 2012
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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。我们研究以下问题的复杂性，对于有理数x和y：给定一个平面图G作为输入，确定T（G;x，y）。Vertigan完全映射了精确计算平面图的Tutte多项式的复杂性。他表明，这个问题可以解决在多项式时间，如果（x，y）是在双曲线H_q由（x-1）（y-1）=q为q=1或q=2或如果（x，y）是其中的两个特殊点（x，y）=（-1，-1）或（x，y）=（1，1）。否则，问题就是#P-hard。在本文中，我们考虑的问题，近似T（G;x，y），在通常意义上的“完全多项式随机逼近计划”或FPRAS。粗略地说，要求FPRAS在多项式时间内以高概率产生具有小的相对误差的答案。假设NP是不同的RP，我们表明，有没有FPRAS的Tutte多项式在很大一部分的（x，y）平面。特别地，如果x&gt;1，y <-1 or if y>1，x&lt;-1或者如果x&lt;0，y <0 and q>5，则不存在FPRAS。此外，如果x&lt;1，y&lt;1且q=3，则不存在FPRAS。对于q&gt;5，我们的结果是有趣的，因为它表明对于任何正的n，在（x，y）=（1-q/（1+ n），-n）处不存在FPRAS，但它留下了极限点n =0，这对应于近似计数平面图的q-着色。

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: given as input a planar graph G, determine T(G;x,y). Vertigan completely mapped the complexity of exactly computing the Tutte polynomial of a planar graph. He showed that the problem can be solved in polynomial time if (x,y) is on the hyperbola H_q given by (x-1)(y-1)=q for q=1 or q=2 or if (x,y) is one of the two special points (x,y)=(-1,-1) or (x,y)=(1,1). Otherwise, the problem is #P-hard. In this paper, we consider the problem of approximating T(G;x,y), in the usual sense of "fully polynomial randomised approximation scheme" or FPRAS. Roughly speaking, an FPRAS is required to produce, in polynomial time and with high probability, an answer that has small relative error. Assuming that NP is different from RP, we show that there is no FPRAS for the Tutte polynomial in a large portion of the (x,y) plane. In particular, there is no FPRAS if x>1, y<-1 or if y>1, x<-1 or if x<0, y<0 and q>5. Also, there is no FPRAS if x<1, y<1 and q=3. For q>5, our result is intriguing because it shows that there is no FPRAS at (x,y)=(1-q/(1+epsilon),-epsilon) for any positive epsilon but it leaves open the limit point epsilon=0, which corresponds to approximately counting q-colourings of a planar graph.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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