图同构方法的一组反例

Constructing Hard Examples for Graph Isomorphism

We describe a method for generating graphs that provide difficult examples for practical Graph Isomorphism testers. We first give the theoretical construction, showing that we can have a family of graphs without any non-trivial automorphisms which also have high Weisfeiler-Leman dimension. The construction is based on properties of random 3XOR-formulas. We describe how to convert such a formula into a graph which has the desired properties with high probability. We validate the method by an experimental implementation. We construct random formulas and validate them with a SAT solver to filter …

A Polynomial Time Algorithm for Graph Isomorphism

We claimed that there is a polynomial algorithm to test if two graphs are isomorphic. But the algorithm is wrong. It only tests if the adjacency matrices of two graphs have the same eigenvalues. There is a counterexample of two non-isomorphic graphs with the same eigenvalues.

Tight Bounds on the Asymptotic Descriptive Complexity of Subgraph Isomorphism

Let $v(F)$ denote the number of vertices in a fixed connected pattern graph $F$. We show an infinite family of patterns $F$ such that the existence of a subgraph isomorphic to $F$ is expressible by a first-order sentence of quantifier depth $\frac23\,v(F)+1$, assuming that the host graph is sufficiently large and connected. On the other hand, this is impossible for any $F$ with using less than $\frac23\,v(F)-2$ first-order variables.

Accelerations for Graph Isomorphism

In this paper, we present two main results. First, by only one conjecture (Conjecture 2.9) for recognizing a vertex symmetric graph, which is the hardest task for our problem, we construct an algorithm for finding an isomorphism between two graphs in polynomial time $ O(n^{3}) $. Second, without that conjecture, we prove the algorithm to be of quasi-polynomial time $ O(n^{1.5\log n}) $. The conjectures in this paper are correct for all graphs of size no larger than $ 5 $ and all graphs we have encountered. At least the conjecture for determining if a graph is vertex symmetric is quite true int…

An exponential lower bound for Individualization-Refinement algorithms for Graph Isomorphism

The individualization-refinement paradigm provides a strong toolbox for testing isomorphism of two graphs and indeed, the currently fastest implementations of isomorphism solvers all follow this approach. While these solvers are fast in practice, from a theoretical point of view, no general lower bounds concerning the worst case complexity of these tools are known. In fact, it is an open question whether individualization-refinement algorithms can achieve upper bounds on the running time similar to the more theoretical techniques based on a group theoretic approach. In this work we give a nega…

关于图同构的量子筛算法的不可能性

On the Impossibility of a Quantum Sieve Algorithm for Graph Isomorphism

众所周知，在隐藏子群问题（HSP）的框架内工作的图同构的任何量子算法必须在Omega（n log n）陪集状态上执行高度纠缠的测量。如何有效地进行这种测量的唯一已知模型之一是用于二面体群中HSP的Kuperberg算法，其中量子态根据张量积分解为不可约表示而自适应地组合和测量。这种“量子筛”从陪集态开始，一路向下，直到概率不同的表示，例如，隐藏的子群是平凡的还是非平凡的。在本文中，我们给出了强有力的证据，没有这样的方法可以成功的图同构。特别地，我们考虑图同构到圈积S_n \wr Z_2上的HSP的自然约化。我们证明，模的对称群的渐近性质的群论猜想，无论我们使用什么规则来自适应组合量子态，有一个常数b > 0，使得在这个家庭中没有算法可以解决图同构在e^{b sqrt{n}}时间。特别地，这样的算法本质上不比最好的已知经典算法好，其运行时间是e^{O（sqrt{n \log n}）}。