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

Complexity of Token Swapping and its Variants

In the Token Swapping problem we are given a graph with a token placed on each vertex. Each token has exactly one destination vertex, and we try to move all the tokens to their destinations, using the minimum number of swaps, i.e., operations of exchanging the tokens on two adjacent vertices. As the main result of this paper, we show that Token Swapping is $W[1]$-hard parameterized by the length $k$ of a shortest sequence of swaps. In fact, we prove that, for any computable function $f$, it cannot be solved in time $f(k)n^{o(k / \log k)}$ where $n$ is the number of vertices of the input graph, unless the ETH fails. This lower bound almost matches the trivial $n^{O(k)}$-time algorithm. We also consider two generalizations of the Token Swapping, namely Colored Token Swapping (where the tokens have different colors and tokens of the same color are indistinguishable), and Subset Token Swapping (where each token has a set of possible destinations). To complement the hardness result, we prove that even the most general variant, Subset Token Swapping, is FPT in nowhere-dense graph classes. Finally, we consider the complexities of all three problems in very restricted classes of graphs: graphs of bounded treewidth and diameter, stars, cliques, and paths, trying to identify the borderlines between polynomial and NP-hard cases.Comment: 23 pages, 7 Figure

A Faster Parameterized Algorithm for Treedepth

The width measure \emph{treedepth}, also known as vertex ranking, centered coloring and elimination tree height, is a well-established notion which has recently seen a resurgence of interest. We present an algorithm which---given as input an $n$-vertex graph, a tree decomposition of the graph of width $w$, and an integer $t$---decides Treedepth, i.e. whether the treedepth of the graph is at most $t$, in time $2^{O(wt)} \cdot n$. If necessary, a witness structure for the treedepth can be constructed in the same running time. In conjunction with previous results we provide a simple algorithm and a fast algorithm which decide treedepth in time $2^{2^{O(t)}} \cdot n$ and $2^{O(t^2)} \cdot n$, respectively, which do not require a tree decomposition as part of their input. The former answers an open question posed by Ossona de Mendez and Nesetril as to whether deciding Treedepth admits an algorithm with a linear running time (for every fixed $t$) that does not rely on Courcelle's Theorem or other heavy machinery. For chordal graphs we can prove a running time of $2^{O(t \log t)}\cdot n$ for the same algorithm.Comment: An extended abstract was published in ICALP 2014, Track

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