On the graph isomorphism problem
Shmuel Friedland
我们将图同构问题与某些非负变量线性方程组的可解性联系起来。这个版本取代了本文的前两个版本。
We relate the graph isomorphism problem to the solvability of certain systems of linear equations with nonnegative variables. This version replaces the two previous versions of this paper.
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The complexity of the graph isomorphism problem for trapezoid graphs has been open over a decade. This paper shows that the problem is GI-complete. More precisely, we show that the graph isomorphism problem is GI-complete for comparability graphs of partially ordered sets with interval dimension 2 and height 3. In contrast, the problem is known to be solvable in polynomial time for comparability graphs of partially ordered sets with interval dimension at most 2 and height at most 2.
A Family of Counter Examples to an Approach to Graph Isomorphism
我们给出一个家庭的反例表明，两个序列的多面体$Φ_{n,n}$和$Ψ_{n,n}$是不同的。最近S. Friedland在图同构的多项式时间算法的尝试。
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 …
Graphs with high symmetry or regularity are the main source for experimentally hard instances of the notoriously difficult graph isomorphism problem. In this paper, we study the computational complexity of isomorphism testing for line graphs of $t$-$(v,k,λ)$ designs. For this class of highly regular graphs, we obtain a worst-case running time of $O(v^{\log v + O(1)})$ for bounded parameters $t,k,λ$. In a first step, our approach makes use of the Babai--Luks algorithm to compute canonical forms of $t$-designs. In a second step, we show that $t$-designs can be reconstructed from their line graph…
We show that a nontrivial graph isomorphism problem of two undirected graphs, and more generally, the permutation similarity of two given $n\times n$ matrices, is equivalent to equalities of volumes of the induced three convex bounded polytopes intersected with a given sequence of balls, centered at the origin with radii $t_i\in (0,\sqrt{n-1})$, where $\{t_i\}$ is an increasing sequence converging to $\sqrt{n-1}$. These polytopes are characterized by $n^2$ inequalities in at most $n^2$ variables. The existence of fpras for computing volumes of convex bodies gives rise to a semi-frpas of order …
An $H$-graph is an intersection graph of connected subgraphs of a suitable subdivision of a fixed graph $H$. Many important classes of graphs, including interval graphs, circular-arc graphs, and chordal graphs, can be expressed as $H$-graphs, and, in particular, every graph is an $H$-graph for a suitable graph $H$. An $H$-graph is called proper if it has a representation where no subgraph properly contains another. We consider the recognition and isomorphism problems for proper $U$-graphs where $U$ is a unicylic graph. We prove that testing whether a graph is a (proper) $U$-graph, for some $U$…
A commonly studied means of parameterizing graph problems is the deletion distance from triviality (Guo et al. 2004), which counts vertices that need to be deleted from a graph to place it in some class for which efficient algorithms are known. In the context of graph isomorphism, we define triviality to mean a graph with maximum degree bounded by a constant, as such graph classes admit polynomial-time isomorphism tests. We generalise deletion distance to a measure we call elimination distance to triviality, based on elimination trees or tree-depth decompositions. We establish that graph canon…
We extend Babai's quasipolynomial-time graph isomorphism test (STOC 2016) and develop a quasipolynomial-time algorithm for the multiple-coset isomorphism problem. The algorithm for the multiple-coset isomorphism problem allows to exploit graph decompositions of the given input graphs within Babai's group-theoretic framework. We use it to develop a graph isomorphism test that runs in time $n^{\operatorname{polylog}(k)}$ where $n$ is the number of vertices and $k$ is the minimum treewidth of the given graphs and $\operatorname{polylog}(k)$ is some polynomial in $\operatorname{log}(k)$. Our resul…
Graph Isomorphism is PSPACE-complete
结合A.R. Meyer和L. J. Stockmeyer“平方正则表达式的等价问题需要指数空间”，以及K.S. Booth“Isomorphism testing for graphs，semigroups，and finite automata are polynomiamly equivalent problems”表明图同构是PSPACE-完全的。
The isomorphism problem is known to be efficiently solvable for interval graphs, while for the larger class of circular-arc graphs its complexity status stays open. We consider the intermediate class of intersection graphs for families of circular arcs that satisfy the Helly property. We solve the isomorphism problem for this class in logarithmic space. If an input graph has a Helly circular-arc model, our algorithm constructs it canonically, which means that the models constructed for isomorphic graphs are equal.
Testing Isomorphism of Graphs in Polynomial Time
本文提出利用Weisfeiler-Lehman算法计算结合图的稳定划分以产生自同构划分，从而在多项式时间内测试图同构，证明该问题属于复杂度类P。
3-connected Planar Graph Isomorphism is in Log-space
证明了3连通平面图的同构可以在确定性对数空间中判定。这改进了先前已知的Thierauf和Wagner的结合UL $\cap$ coUL。
The Graph Isomorphism problem restricted to graphs of bounded treewidth or bounded tree distance width are known to be solvable in polynomial time [Bod90],[YBFT99]. We give restricted space algorithms for these problems proving the following results: - Isomorphism for bounded tree distance width graphs is in L and thus complete for the class. We also show that for this kind of graphs a canon can be computed within logspace. - For bounded treewidth graphs, when both input graphs are given together with a tree decomposition, the problem of whether there is an isomorphism which respects the decom…
We give a $O^*(k^{O(k)})$ time isomorphism testing algorithm for graphs of eigenvalue multiplicity bounded by $k$ which improves on the previous best running time bound of $O^*(2^{O(k^2/\log k)})$.
In this paper we resolve the complexity of the isomorphism problem on all but finitely many of the graph classes characterized by two forbidden induced subgraphs. To this end we develop new techniques applicable for the structural and algorithmic analysis of graphs. First, we develop a methodology to show isomorphism completeness of the isomorphism problem on graph classes by providing a general framework unifying various reduction techniques. Second, we generalize the concept of the modular decomposition to colored graphs, allowing for non-standard decompositions. We show that, given a suitab…
We study the complexity of the Graph Isomorphism problem on graph classes that are characterized by a finite number of forbidden induced subgraphs, focusing mostly on the case of two forbidden subgraphs. We show hardness results and develop techniques for the structural analysis of such graph classes, which applied to the case of two forbidden subgraphs give the following results: A dichotomy into isomorphism complete and polynomial-time solvable graph classes for all but finitely many cases, whenever neither of the forbidden graphs is a clique, a pan, or a complement of these graphs. Further …
On the Complexity of Matroid Isomorphism Problem
研究了两个拟阵同构的判定复杂性。很容易看出问题出在$Σ_2^p$中。在线性拟阵的情况下，这是多项式增长领域表示，我们注意到，这个问题是不太可能是$Σ_2^p$ -完全的，是$\co\NP$ -硬。我们证明了当拟阵的秩有界时，线性拟阵同构和拟阵同构都是图同构的多项式时间多-一等价。给出了图拟阵同构问题到图同构问题的多项式时间Turing约简。利用这一点，我们能够证明，平面图的图形拟阵同构测试可以在确定性多项式时间内完成。然后给出了有界秩拟阵同构问题到图拟阵同构的多项式时间多一约化，从而证明了上述问题是多项式时间等价的.进一步，对于线性拟阵和图拟阵，我们证明了自同构问题与相应的同构问题是多项式时间等价的。此外，我们还给出了图拟阵的自同构群的多项式时间隶属度检验算法。
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.
On the Solution of Graph Isomorphism by Dynamical Algorithms
近年来，一些多项式算法的动态性质已被提出来解决图同构问题。在本文中，我们提出了一个推广的方法暴露在cond-mat/0209112，并发现，这种动态算法所涵盖的组合方法。这是可能的推断，多项式动态算法解决图同构涵盖适当的多项式组合方法，从而解决了同样的弱点，因为最后的。
It is well-known [KST93] that the complexity of the Graph Automorphism problem is characterized by a special case of Graph Isomorphism, where the input graphs satisfy the "promise" of being rigid (that is, having no nontrivial automorphisms). In this brief note, we observe that the reduction of Graph Automorphism to the Rigid Graph Ismorphism problem can be accomplished even using Grollman and Selman's notion of a "smart reduction".
Babel has shown that for an extended class of chordal (6, 3) graphs the coarsest regular simplicial partition is equivalent to the graph's automorphism partition. We give a reversible transformation for any graph to one of these graph by using Booth's reduction of a graph to a chordal graph and elimination of Babel's forbidden subgraphs for these graphs by adding edges to them.
Two graph isomorphism polytopes
最近，在图同构方面考虑了某些$(n!)^2$张量的凸包$ψ_{n,n}$。我们考虑这些张量中的$n!$对角线的凸包$ψ_n$。我们显示：1.多面体$ψ_n$是$ψ_{n,n}$的一个面。2.判定图$G$是否具有与$H$同构的子图简化为对$ψ_n$的优化。3.对$ψ_n$的优化简化为对$ψ_{n,n}$的优化。特别地，这意味着子图同构问题简化为在$ψ_{n,n}$上的优化。
The clique-width is a measure of complexity of decomposing graphs into certain tree-like structures. The class of graphs with bounded clique-width contains bounded tree-width graphs. We give a polynomial time graph isomorphism algorithm for graphs with clique-width at most three. Our work is independent of the work by Grohe et al. \cite{grohe2015isomorphism} showing that the isomorphism problem for graphs of bounded clique-width is polynomial time.
Computing Complete Graph Isomorphisms and Hamiltonian Cycles from Partial Ones
我们证明，计算一对顶点映射到彼此之间的同构$ϕ$两个同构图是很难计算$ϕ$本身。这一结果最佳改善后，结果的Gál等人。我们建立了一个类似的，虽然稍弱，结果计算完全的图从部分哈密尔顿圈。
In this paper we show lower bounds for a certain large class of algorithms solving the Graph Isomorphism problem, even on expander graph instances. Spielman [25] shows an algorithm for isomorphism of strongly regular expander graphs that runs in time exp(O(n^(1/3)) (this bound was recently improved to expf O(n^(1/5) [5]). It has since been an open question to remove the requirement that the graph be strongly regular. Recent algorithmic results show that for many problems the Lasserre hierarchy works surprisingly well when the underlying graph has expansion properties. Moreover, recent work of …
The Isomorphism Problem for Planar 3-Connected Graphs is in Unambiguous Logspace
平面图的同构问题是一个有效的可解问题。对于平面3-连通图，同构问题可以用有效的并行算法来解决，它属于$AC^1$类。本文将平面3-连通图的上界推广到无二义性的对数空间，实际上推广到$UL \cap coUL$。作为我们的方法的结果，我们得到了有向图的同构问题是在$NL$。我们还表明，问题是困难的$L$。
A message-passing procedure for solving the graph isomorphism problem is proposed. The procedure resembles the belief-propagation algorithm in the context of graphical models inference and LDPC decoding. To enable the algorithm, the input graphs are transformed into intermediate canonical representations of bipartite graphs. The matching procedure injects specially designed input patterns to the canonical graphs and runs a message-passing algorithm to generate two output fingerprints that are matched if and only if the input graphs are isomorphic.
Testing Graph Isomorphism in Parallel by Playing a Game
我们的出发点是观察到，如果在一个类C中的图具有低描述的复杂性在一阶逻辑，那么同构问题C是可解决的快速并行算法（本质上，由一个简单的组合算法称为多维Weisfeiler-Lehman算法）。使用这种方法，我们证明了有界树宽的图的同构是可测试的TC 1，回答了一个开放的问题所提出的Escherasekharan。此外，我们得到了一个AC 1算法测试同构的旋转系统（组合规格的图嵌入）。AC 1上界以前是已知的，但这个界可以通过简单的Weisfeiler-Lehman算法实现的事实是新的。结合已有的结果，给出了平面图的一个新的AC 1同构算法。
Planar Graphs: Logical Complexity and Parallel Isomorphism Tests
我们证明了每一个三连通平面图是可定义的一阶句子，使用最多15个变量，量词深度最多$11\log_2 n+43$。因此，这样的图的规范形式可以通过14维Weisfeiler-Lehman算法在$AC^1$中计算。这提供了另一种方法来证明平面图同构在$AC^1$中是可解的。
The Isomorphism Problem of Power Graphs and a Question of Cameron
本文研究有限幂零群产生的幂图、有向幂图和增强幂图的同构问题，给出无需底层群的多项式时间算法，并解决了 Cameron 关于从幂图或增强幂图恢复有向幂图的问题。
Graph canonization is the problem of computing a unique representative, a canon, from the isomorphism class of a given graph. This implies that two graphs are isomorphic exactly if their canons are equal. We show that graphs of bounded tree width can be canonized by logarithmic-space (logspace) algorithms. This implies that the isomorphism problem for graphs of bounded tree width can be decided in logspace. In the light of isomorphism for trees being hard for the complexity class logspace, this makes the ubiquitous class of graphs of bounded tree width one of the few classes of graphs for whic…
Dynamic Planar Graph Isomorphism is in DynFO
本文研究动态平面图同构性问题，证明其可通过一阶逻辑公式和多项式大小辅助数据维护，属于DynFO复杂性类，并提出常数时间并行算法。
Graph Isomorphism is such an important problem in computer science, that it has been widely studied over the last decades. It is well known that it belongs to NP class, but is not NP-complete. It is thought to be of comparable difficulty to integer factorisation. The best known proved algorithm to solve this problem in general, was proposed by László Babai and Eugene Luks in 1983. Recently, there has been some research in the topic by using quantum computing, that also leads the present piece of research. In fact, we present a quantum computing algorithm that defines an invariant over Graph Is…
The presented material is devoted to the equivalent conversion from the vertex graphs to the edge graphs. We suggest that the proved theorems solve the problem of the isomorphism of graphs, the problem of the graph's enumeration with the help of the effective algorithms without their preliminary plotting, etc. The examining of the transformation of the vertex graphs into the edge graph and the opposite operation illustrates the reasons of the appearance of the NP-completeness from the point of view of the graph theory. We suggest that it also illustrates the synchronous possibility and impossi…
The graph isomorphism problem is theoretically interesting and also has many practical applications. The best known classical algorithms for graph isomorphism all run in time super-polynomial in the size of the graph in the worst case. An interesting open problem is whether quantum computers can solve the graph isomorphism problem in polynomial time. In this paper, an algorithm is shown which can decide if two rooted trees are isomorphic in polynomial time. Although this problem is easy to solve efficiently on a classical computer, the techniques developed may be useful as a basis for quantum …
Building on work of Cai, Fürer, and Immerman \cite{CFI92}, we show two hardness results for the Graph Isomorphism problem. First, we show that there are pairs of nonisomorphic $n$-vertex graphs $G$ and $H$ such that any sum-of-squares (SOS) proof of nonisomorphism requires degree $Ω(n)$. In other words, we show an $Ω(n)$-round integrality gap for the Lasserre SDP relaxation. In fact, we show this for pairs $G$ and $H$ which are not even $(1-10^{-14})$-isomorphic. (Here we say that two $n$-vertex, $m$-edge graphs $G$ and $H$ are $α$-isomorphic if there is a bijection between their vertices whic…
It is well-known that the graph isomorphism problem can be posed as an equivalent problem of determining whether an auxiliary graph structure contains a clique of specific order. However, the algorithms that have been developed so far for this problem are either not efficient or not exact. In this paper, we present a new algorithm which solves this equivalent formulation via semidefinite programming. Specifically, we show that the problem of determining whether the auxiliary graph contains a clique of specific order can be formulated as a semidefinite programming problem, and can thus be (almo…
Combinatorial invariants for graph isomorphism problem
提出的方法在多项式时间内计算每个顶点的大量不变量，这些不变量不随图的同构而改变，并且应该完全确定图。例如，给定起始点的长度为k的闭合路径的数量，可以认为是邻接矩阵的k次幂的对角项。当k=2时，我们得到的是实点的度不变，越高，描述的局部拓扑就越深。现在，如果两个图是同构的，它们有相同的一组这样的不变量向量-我们可以对这些向量进行字典排序并比较它们。如果它们一致，排序的置换允许重构同构。我提出的论点是，这些不变量应该完全决定图，但不幸的是，我不能证明这一点在这一刻。这种方法可以带来希望，也许P=NP -而不是检查所有的实例，我们应该对这些大数字进行算术。
We give a fixed-parameter tractable algorithm that, given a parameter $k$ and two graphs $G_1,G_2$, either concludes that one of these graphs has treewidth at least $k$, or determines whether $G_1$ and $G_2$ are isomorphic. The running time of the algorithm on an $n$-vertex graph is $2^{O(k^5\log k)}\cdot n^5$, and this is the first fixed-parameter algorithm for Graph Isomorphism parameterized by treewidth. Our algorithm in fact solves the more general canonization problem. We namely design a procedure working in $2^{O(k^5\log k)}\cdot n^5$ time that, for a given graph $G$ on $n$ vertices, eit…
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…
It is unknown whether two graphs can be tested for isomorphism in polynomial time. A classical approach to the Graph Isomorphism Problem is the d-dimensional Weisfeiler-Lehman algorithm. The d-dimensional WL-algorithm can distinguish many pairs of graphs, but the pairs of non-isomorphic graphs constructed by Cai, Furer and Immerman it cannot distinguish. If d is fixed, then the WL-algorithm runs in polynomial time. We will formulate the Graph Isomorphism Problem as an Orbit Problem: Given a representation V of an algebraic group G and two elements v_1,v_2 in V, decide whether v_1 and v_2 lie i…
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}）}。
In this paper, we study the parallel and the space complexity of the graph isomorphism problem (\GI{}) for several parameterizations. Let $\mathcal{H}=\{H_1,H_2,\cdots,H_l\}$ be a finite set of graphs where $|V(H_i)|\leq d$ for all $i$ and for some constant $d$. Let $\mathcal{G}$ be an $\mathcal{H}$-free graph class i.e., none of the graphs $G\in \mathcal{G}$ contain any $H \in \mathcal{H}$ as an induced subgraph. We show that \GI{} parameterized by vertex deletion distance to $\mathcal{G}$ is in a parameterized version of $\AC^1$, denoted $\PL$-$\AC^1$, provided the colored graph isomorphism …
In this paper, we prove results on the relationship between the complexity of the group and color isomorphism problems. The difficulty of color isomorphism problems is known to be closely linked to the the composition factors of the permutation group involved. Previous works are primarily concerned with applying color isomorphism to bou nded degree graph isomorphism, and have therefore focused on the alternating composit ion factors, since those are the bottleneck in the case of graph isomorphism. We consider the color isomorphism problem with composition factors restricted to those other than…
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