[论文解读] A solution to Lov\'asz's Seventeenth problem

Hamed Hatami, Serguei Norine|arXiv (Cornell University)|May 13, 2010

Graph theory and applications被引用 1

一句话总结

本文通过从实变量多项式进行约化，解决了洛瓦兹的第十七个问题，证明并非所有正量子图（即图同态密度之间的不等式）都可以表示为平方和，从而表明在量子图设定下，希尔伯特第十七问题的阿廷解法的类比不成立。本文还证明了验证此类不等式的不可判定性。

ABSTRACT

The purpose of this article is to show that even the most elementary problems in asymptotic extremal graph theory can be highly non-trivial. We study linear inequalities between graph homomorphism densities. In the language of quantum graphs the validity of such an inequality is equivalent to the positivity of a corresponding quantum graph. Similar to the setting of polynomials, a quantum graph that can be represented as a sum of squares of labeled quantum graphs is necessarily positive. Lovasz asks whether the opposite is also true. We answer this question and also a related question of Razborov in the negative by introducing explicit valid inequalities that do not satisfy the required conditions. Our solution to these problems is based on a reduction from real multivariate polynomials and uses the fact that there are positive polynomials that cannot be expressed as sums of squares of polynomials. It is known that the problem of determining whether a multivariate polynomial is positive is decidable. Hence it is very natural to ask Is the problem of determining the validity of a linear inequality between homomorphism densities decidable? We give a negative answer to this question which shows that such inequalities are inherently difficult in their full generality. Furthermore we deduce from this fact that the analogue of Artin's solution to Hilbert's seventeenth problem does not hold in the setting of quantum graphs.

研究动机与目标

• 解决洛瓦兹关于所有正量子图是否可表示为平方和的问题。
• 研究图同态密度之间线性不等式可判定性的性质。
• 确定希尔伯特第十七问题的阿廷解法在量子图语境下的类比是否成立。
• 为洛瓦兹和拉兹博罗夫关于正量子图表示形式的相关问题提供否定答案。

提出的方法

• 从实变量多项式约化到量子图，利用已知结果：存在不能表示为平方和的正多项式。
• 构造明确的反例，以反驳所有正量子图均为平方和的猜想。
• 利用实代数几何中已知的结论：存在不能表示为平方和的正多项式。
• 建立量子图正性与非负实变量多项式之间的对应关系。
• 证明判断图同态密度之间线性不等式是否有效的决策问题是不可判定的。
• 运用逻辑与代数技术，证明此类不等式不存在通用的算法解法。

实验结果

研究问题

• RQ1 每个正量子图是否都能表示为标记量子图的平方和？
• RQ2 判断图同态密度之间线性不等式是否有效的决策问题是否可判定？
• RQ3 希尔伯特第十七问题的阿廷解法在量子图设定下是否成立？
• RQ4 是否存在无法表示为平方和的图同态密度之间的有效不等式？
• RQ5 在渐近极值图论中，验证不等式的逻辑复杂度是什么？

主要发现

• 并非所有正量子图都能表示为标记量子图的平方和，从而为洛瓦兹的第十七个问题提供了否定答案。
• 存在明确有效的图同态密度之间不等式，但不满足平方和条件。
• 判断图同态密度之间线性不等式是否有效的决策问题是不可判定的。
• 希尔伯特第十七问题的阿廷解法在量子图设定下不成立。
• 不可判定性结果源于将问题约化为判断多元多项式是否非负的不可判定性问题。