Abstract
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 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.
Highlights
In reconfiguration problems, we are interested to transform a combinatorial or geometric object from one state to another, by performing a sequence of simple operations
We prove that Subset Token Swapping is FPT in k +, where k is the number of allowed swaps, and is the maximum degree of the input graph
Since after fixing the destinations we obtain an instance of Token Swapping, which is polynomially solvable on paths, we observe that each feasible solution s for I corresponds to a perfect matching in G
Summary
We are interested to transform a combinatorial or geometric object from one state to another, by performing a sequence of simple operations. Assuming the ETH, for any computable function f , Token Swapping cannot be solved in time f (k)(n + m)o(k/ log k) where n and m are respectively the number of vertices and edges of the input graph. Observe that this lower bound shows that the simple nO(k)-time algorithm is almost best possible. We use this meta-theorem to show the existence of an FPT time algorithm for Subset Token Swapping, restricted to nowhere-dense graph classes It is always accompanied by an appropriate reference to Theorem/Proposition “?” Denotes unknown cases
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
