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 negative answer to this question and construct a family of graphs on which algorithms based on the individualization-refinement paradigm require exponential time. Contrary to a previous construction of Miyazaki, that only applies to a specific implementation within the individualization-refinement framework, our construction is immune to changing the cell selector, or adding various heuristic invariants to the algorithm. Furthermore, our graphs also provide exponential lower bounds in the case when the $k$-dimensional Weisfeiler-Leman algorithm is used to replace the standard color refinement operator and the arguments even work when the entire automorphism group of the inputs is initially provided to the algorithm.Comment: 21 page

Scott : A method for representing graphs asrooted trees for graph canonization

International audienceGraphs increasingly stand out as an essential data structurein the field of data sciences. To study graphs, or sub-graphs, that char-acterize a set of observations, it is necessary to describe them formally,in order to characterize equivalence relations that make sense in thescope of the considered application domain. Hence we seek to define acanonical graph notation, so that two isomorphic (sub) graphs have thesame canonical form. Such notation could subsequently be used to indexand retrieve graphs or to embed them efficiently in some metric space.Sequential optimized algorithms solving this problem exist, but do notdeal with labeled edges, a situation that occurs in important applicationdomains such as chemistry. We present in this article a new algorithmbased on graph rewriting that provides a general and complete solution tothe graph canonization problem. Although not reported here, the formalproof of the validity of our algorithm has been established. This claim isclearly supported empirically by our experimentation on synthetic com-binatorics as well as natural graphs. Furthermore, our algorithm supportsdistributed implementations, leading to efficient computing perspectives

The Structure of Level-k Phylogenetic Networks

International audienceEvolution is usually described as a phylogenetic tree, but due to some exchange of genetic material, it can be represented as a phylogenetic network which has an underlying tree structure. The notion of level was recently introduced as a parameter on realistic kinds of phylogenetic networks to express their complexity and tree-likeness. We study the structure of level-k networks, and how they can be decomposed into level-k generators. We also provide a polynomial time algorithm which takes as input the set of level-k generators and builds the set of level-(k+1) generators. Finally, with a simulation study, we evaluate the proportion of level-k phylogenetic networks among networks generated according to the coalescent model with recombination