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

Towards Resistance Sparsifiers

We study resistance sparsification of graphs, in which the goal is to find a sparse subgraph (with reweighted edges) that approximately preserves the effective resistances between every pair of nodes. We show that every dense regular expander admits a $(1+\epsilon)$-resistance sparsifier of size $\tilde O(n/\epsilon)$, and conjecture this bound holds for all graphs on $n$ nodes. In comparison, spectral sparsification is a strictly stronger notion and requires $\Omega(n/\epsilon^2)$ edges even on the complete graph. Our approach leads to the following structural question on graphs: Does every dense regular expander contain a sparse regular expander as a subgraph? Our main technical contribution, which may of independent interest, is a positive answer to this question in a certain setting of parameters. Combining this with a recent result of von Luxburg, Radl, and Hein~(JMLR, 2014) leads to the aforementioned resistance sparsifiers

Distributed CONGEST Algorithm for Finding Hamiltonian Paths in Dirac Graphs and Generalizations

We study the problem of finding a Hamiltonian cycle under the promise that the input graph has a minimum degree of at least $n/2$, where $n$ denotes the number of vertices in the graph. The classical theorem of Dirac states that such graphs (a.k.a. Dirac graphs) are Hamiltonian, i.e., contain a Hamiltonian cycle. Moreover, finding a Hamiltonian cycle in Dirac graphs can be done in polynomial time in the classical centralized model. This paper presents a randomized distributed CONGEST algorithm that finds w.h.p. a Hamiltonian cycle (as well as maximum matching) within $O(\log n)$ rounds under the promise that the input graph is a Dirac graph. This upper bound is in contrast to general graphs in which both the decision and search variants of Hamiltonicity require $\tilde{\Omega}(n^2)$ rounds, as shown by Bachrach et al. [PODC'19]. In addition, we consider two generalizations of Dirac graphs: Ore graphs and Rahman-Kaykobad graphs [IPL'05]. In Ore graphs, the sum of the degrees of every pair of non-adjacent vertices is at least $n$, and in Rahman-Kaykobad graphs, the sum of the degrees of every pair of non-adjacent vertices plus their distance is at least $n+1$. We show how our algorithm for Dirac graphs can be adapted to work for these more general families of graphs

Hamilton cycles in hypergraphs below the Dirac threshold

We establish a precise characterisation of $4$-uniform hypergraphs with minimum codegree close to $n/2$ which contain a Hamilton $2$-cycle. As an immediate corollary we identify the exact Dirac threshold for Hamilton $2$-cycles in $4$-uniform hypergraphs. Moreover, by derandomising the proof of our characterisation we provide a polynomial-time algorithm which, given a $4$-uniform hypergraph $H$ with minimum codegree close to $n/2$, either finds a Hamilton $2$-cycle in $H$ or provides a certificate that no such cycle exists. This surprising result stands in contrast to the graph setting, in which below the Dirac threshold it is NP-hard to determine if a graph is Hamiltonian. We also consider tight Hamilton cycles in $k$-uniform hypergraphs $H$ for $k \geq 3$, giving a series of reductions to show that it is NP-hard to determine whether a $k$-uniform hypergraph $H$ with minimum degree $\delta(H) \geq \frac{1}{2}|V(H)| - O(1)$ contains a tight Hamilton cycle. It is therefore unlikely that a similar characterisation can be obtained for tight Hamilton cycles.Comment: v2: minor revisions in response to reviewer comments, most pseudocode and details of the polynomial time reduction moved to the appendix which will not appear in the printed version of the paper. To appear in Journal of Combinatorial Theory, Series