The Graph Isomorphism Problem and approximate categories

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 in the same G-orbit. Then we attack the Orbit Problem by constructing certain approximate categories C_d(V), d=1,2,3,... whose objects include the elements of V. We show that v_1 and v_2 are not in the same orbit by showing that they are not isomorphic in the category C_d(V) for some d. For every d this gives us an algorithm for isomorphism testing. We will show that the WL-algorithms reduce to our algorithms, but that our algorithms cannot be reduced to the WL-algorithms. Unlike the Weisfeiler-Lehman algorithm, our algorithm can distinguish the Cai-Furer-Immerman graphs in polynomial time.Comment: 29 page

Limitations of Algebraic Approaches to Graph Isomorphism Testing

We investigate the power of graph isomorphism algorithms based on algebraic reasoning techniques like Gr\"obner basis computation. The idea of these algorithms is to encode two graphs into a system of equations that are satisfiable if and only if if the graphs are isomorphic, and then to (try to) decide satisfiability of the system using, for example, the Gr\"obner basis algorithm. In some cases this can be done in polynomial time, in particular, if the equations admit a bounded degree refutation in an algebraic proof systems such as Nullstellensatz or polynomial calculus. We prove linear lower bounds on the polynomial calculus degree over all fields of characteristic different from 2 and also linear lower bounds for the degree of Positivstellensatz calculus derivations. We compare this approach to recently studied linear and semidefinite programming approaches to isomorphism testing, which are known to be related to the combinatorial Weisfeiler-Lehman algorithm. We exactly characterise the power of the Weisfeiler-Lehman algorithm in terms of an algebraic proof system that lies between degree-k Nullstellensatz and degree-k polynomial calculus

Computational complexity of reconstruction and isomorphism testing for designs and line graphs

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 $t$-$(v,k,\lambda)$ 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 $t,k,\lambda$. 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 graphs in polynomial-time. The first is algebraic in nature, the second purely combinatorial. For both, profound structural knowledge in design theory is required. Our results extend earlier complexity results about isomorphism testing of graphs generated from Steiner triple systems and block designs.Comment: 12 pages; to appear in: "Journal of Combinatorial Theory, Series A

The Module Isomorphism Problem Reconsidered

Algorithms to decide isomorphism of modules have been honed continually over the last 30 years, and their range of applicability has been extended to include modules over a wide range of rings. Highly efficient computer implementations of these algorithms form the bedrock of systems such as GAP and MAGMA, at least in regard to computations with groups and algebras. By contrast, the fundamental problem of testing for isomorphism between other types of algebraic structures -- such as groups, and almost any type of algebra -- seems today as intractable as ever. What explains the vastly different complexity status of the module isomorphism problem? This paper argues that the apparent discrepancy is explained by nomenclature. Current algorithms to solve module isomorphism, while efficient and immensely useful, are actually solving a highly constrained version of the problem. We report that module isomorphism in its general form is as hard as algebra isomorphism and graph isomorphism, both well-studied problems that are widely regarded as difficult. On a more positive note, for cyclic rings we describe a polynomial-time algorithm for the general module isomorphism problem. We also report on a MAGMA implementation of our algorithm

Polynomial-time Isomorphism Test for Groups with Abelian Sylow Towers

We consider the problem of testing isomorphism of groups of order n given by Cayley tables. The trivial n^{log n} bound on the time complexity for the general case has not been improved over the past four decades. Recently, Babai et al. (following Babai et al. in SODA 2011) presented a polynomial-time algorithm for groups without abelian normal subgroups, which suggests solvable groups as the hard case for group isomorphism problem. Extending recent work by Le Gall (STACS 2009) and Qiao et al. (STACS 2011), in this paper we design a polynomial-time algorithm to test isomorphism for the largest class of solvable groups yet, namely groups with abelian Sylow towers, defined as follows. A group G is said to possess a Sylow tower, if there exists a normal series where each quotient is isomorphic to Sylow subgroup of G. A group has an abelian Sylow tower if it has a Sylow tower and all its Sylow subgroups are abelian. In fact, we are able to compute the coset of isomorphisms of groups formed as coprime extensions of an abelian group, by a group whose automorphism group is known. The mathematical tools required include representation theory, Wedderburn\u27s theorem on semisimple algebras, and M.E. Harris\u27s 1980 work on p\u27-automorphisms of abelian p-groups. We use tools from the theory of permutation group algorithms, and develop an algorithm for a parameterized versin of the graph-isomorphism-hard setwise stabilizer problem, which may be of independent interest

Cerca d’automorfismes i d’isomorfismes de grafs: algorismes i implementació

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2013, Director: Antoni BensenyGraph Isomorphism problem (GIP) consists in constructing an efficient algorithm for testing whether two graphs are isomorphics and to find, if it exists, one isomorphism,i.e, to find a one-to-one, onto mapping from the set of vertexs of one graph to the set of vertexs of the other graph, so that the edges remain the same. GIP is one of the most interesting problems in Discrete Mathematics, it is essential for Graph Theory as well as for computational complexity theory. The problem has interest in the theoretical side, because it is not known if it is a P or NP-complete problem. On the one hand it is known that many classes of graphs admit polynomial time algorithms for isomorphism testing, such as rooted trees, planar graphs, circular graphs, graphs with bounded vertex degree, etc. On the other hand, the search for isomorphism complete problems has produced a large list of problems, such as bipartite graph isomorphism, regular graph isomorphism, complement graph isomorphism, automorphism orbits search, automorphism generators search, etc. So, GIP is a good candidate for an intermediate computational status between P and NP problems. From the practical point of view, GIP has many applica tions for science and technology: pattern recognition, data mining, computer vision, chemistry and VLSI layout validation. During the last decades many algorithms have been born to solve the problem, most of them based on Canonical labeling and direct backtracking, but Brendan McKay’s Nauty algorithm has stood up among the others because its fine performing. In 2011, José Luis López Presa et al. tested Nauty with some special graphs, Miyazaki’s graphs, and confirmed that it was very slow to find isomorphisms. So, they developed a new algorithm, Conauto, which attacks the problem in an original way, using special techniques to prune the automorphisms searching tree, to become one of the fastest algorithms today. The aim of this work is, on the one hand, to study and analyze the theoretical background the algorithm Conauto relies on and, on the other hand, to try to perform any improvement in some aspects: it has been implemented a graphical interface to visualize the partitions on the graphs and it has been performed two different algorithms to calculate the whole automorphism group of the graph starting from a set of generators; a first one based on brute force (strength) and a second one based on the Schreier-Sims algorithm. An algoritm that calculates all isomorphisms between two graphs, starting from one previous isomorphism and the whole automorphism group of one of the graphs, have also been implemented