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

Counting Euler Tours in Undirected Bounded Treewidth Graphs

We show that counting Euler tours in undirected bounded tree-width graphs is tractable even in parallel - by proving a $\#SAC^1$ upper bound. This is in stark contrast to #P-completeness of the same problem in general graphs. Our main technical contribution is to show how (an instance of) dynamic programming on bounded \emph{clique-width} graphs can be performed efficiently in parallel. Thus we show that the sequential result of Espelage, Gurski and Wanke for efficiently computing Hamiltonian paths in bounded clique-width graphs can be adapted in the parallel setting to count the number of Hamiltonian paths which in turn is a tool for counting the number of Euler tours in bounded tree-width graphs. Our technique also yields parallel algorithms for counting longest paths and bipartite perfect matchings in bounded-clique width graphs. While establishing that counting Euler tours in bounded tree-width graphs can be computed by non-uniform monotone arithmetic circuits of polynomial degree (which characterize $\#SAC^1$) is relatively easy, establishing a uniform $\#SAC^1$ bound needs a careful use of polynomial interpolation.Comment: 17 pages; There was an error in the proof of the GapL upper bound claimed in the previous version which has been subsequently remove