Restricted Space Algorithms for Isomorphism on Bounded Treewidth Graphs

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 decompositions (i.e. considering only isomorphisms mapping bags in one decomposition blockwise onto bags in the other decomposition) is in L. - For bounded treewidth graphs, when one of the input graphs is given with a tree decomposition the isomorphism problem is in LogCFL. - As a corollary the isomorphism problem for bounded treewidth graphs is in LogCFL. This improves the known TC1 upper bound for the problem given by Grohe and Verbitsky [GroVer06].Comment: STACS conference 2010, 12 page

Approximately Counting Embeddings into Random Graphs

Let H be a graph, and let C_H(G) be the number of (subgraph isomorphic) copies of H contained in a graph G. We investigate the fundamental problem of estimating C_H(G). Previous results cover only a few specific instances of this general problem, for example, the case when H has degree at most one (monomer-dimer problem). In this paper, we present the first general subcase of the subgraph isomorphism counting problem which is almost always efficiently approximable. The results rely on a new graph decomposition technique. Informally, the decomposition is a labeling of the vertices such that every edge is between vertices with different labels and for every vertex all neighbors with a higher label have identical labels. The labeling implicitly generates a sequence of bipartite graphs which permits us to break the problem of counting embeddings of large subgraphs into that of counting embeddings of small subgraphs. Using this method, we present a simple randomized algorithm for the counting problem. For all decomposable graphs H and all graphs G, the algorithm is an unbiased estimator. Furthermore, for all graphs H having a decomposition where each of the bipartite graphs generated is small and almost all graphs G, the algorithm is a fully polynomial randomized approximation scheme. We show that the graph classes of H for which we obtain a fully polynomial randomized approximation scheme for almost all G includes graphs of degree at most two, bounded-degree forests, bounded-length grid graphs, subdivision of bounded-degree graphs, and major subclasses of outerplanar graphs, series-parallel graphs and planar graphs, whereas unbounded-length grid graphs are excluded.Comment: Earlier version appeared in Random 2008. Fixed an typo in Definition 3.

A correspondence between rooted planar maps and normal planar lambda terms

A rooted planar map is a connected graph embedded in the 2-sphere, with one edge marked and assigned an orientation. A term of the pure lambda calculus is said to be linear if every variable is used exactly once, normal if it contains no beta-redexes, and planar if it is linear and the use of variables moreover follows a deterministic stack discipline. We begin by showing that the sequence counting normal planar lambda terms by a natural notion of size coincides with the sequence (originally computed by Tutte) counting rooted planar maps by number of edges. Next, we explain how to apply the machinery of string diagrams to derive a graphical language for normal planar lambda terms, extracted from the semantics of linear lambda calculus in symmetric monoidal closed categories equipped with a linear reflexive object or a linear reflexive pair. Finally, our main result is a size-preserving bijection between rooted planar maps and normal planar lambda terms, which we establish by explaining how Tutte decomposition of rooted planar maps (into vertex maps, maps with an isthmic root, and maps with a non-isthmic root) may be naturally replayed in linear lambda calculus, as certain surgeries on the string diagrams of normal planar lambda terms.Comment: Corrected title field in metadat

Minimum cycle and homology bases of surface embedded graphs

We study the problems of finding a minimum cycle basis (a minimum weight set of cycles that form a basis for the cycle space) and a minimum homology basis (a minimum weight set of cycles that generates the $1$-dimensional ($\mathbb{Z}_2$)-homology classes) of an undirected graph embedded on a surface. The problems are closely related, because the minimum cycle basis of a graph contains its minimum homology basis, and the minimum homology basis of the $1$-skeleton of any graph is exactly its minimum cycle basis. For the minimum cycle basis problem, we give a deterministic $O(n^\omega+2^{2g}n^2+m)$-time algorithm for graphs embedded on an orientable surface of genus $g$. The best known existing algorithms for surface embedded graphs are those for general graphs: an $O(m^\omega)$ time Monte Carlo algorithm and a deterministic $O(nm^2/\log n + n^2 m)$ time algorithm. For the minimum homology basis problem, we give a deterministic $O((g+b)^3 n \log n + m)$-time algorithm for graphs embedded on an orientable or non-orientable surface of genus $g$ with $b$ boundary components, assuming shortest paths are unique, improving on existing algorithms for many values of $g$ and $n$. The assumption of unique shortest paths can be avoided with high probability using randomization or deterministically by increasing the running time of the homology basis algorithm by a factor of $O(\log n)$.Comment: A preliminary version of this work was presented at the 32nd Annual International Symposium on Computational Geometr

Canonizing Graphs of Bounded Tree Width in Logspace

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 which the complexity of the isomorphism problem has been exactly determined.Comment: 26 page

Distributed Testing of Graph Isomorphism in the CONGEST Model

In this paper we study the problem of testing graph isomorphism (GI) in the CONGEST distributed model. In this setting we test whether the distributive network, $G_U$, is isomorphic to $G_K$ which is given as an input to all the nodes in the network, or alternatively, only to a single node. We first consider the decision variant of the problem in which the algorithm distinguishes $G_U$ and $G_K$ which are isomorphic from $G_U$ and $G_K$ which are not isomorphic. We provide a randomized algorithm with $O(n)$ rounds for the setting in which $G_K$ is given only to a single node. We prove that for this setting the number of rounds of any deterministic algorithm is $\tilde{\Omega}(n^2)$ rounds, where $n$ denotes the number of nodes, which implies a separation between the randomized and the deterministic complexities of deciding GI. We then consider the \emph{property testing} variant of the problem, where the algorithm is only required to distinguish the case that $G_U$ and $G_K$ are isomorphic from the case that $G_U$ and $G_K$ are \emph{far} from being isomorphic (according to some predetermined distance measure). We show that every algorithm requires $\Omega(D)$ rounds, where $D$ denotes the diameter of the network. This lower bound holds even if all the nodes are given $G_K$ as an input, and even if the message size is unbounded. We provide a randomized algorithm with an almost matching round complexity of $O(D+(\epsilon^{-1}\log n)^2)$ rounds that is suitable for dense graphs. We also show that with the same number of rounds it is possible that each node outputs its mapping according to a bijection which is an \emph{approximated} isomorphism. We conclude with simple simulation arguments that allow us to obtain essentially tight algorithms with round complexity $\tilde{O}(D)$ for special families of sparse graphs