Complexity of Token Swapping and Its Variants

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 timewhere 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.

Introduction to Reconfiguration

“…In "parallel" variants, swapping is allowed on non-adjacent edges at the same time; the problem of determining the minimum number of swaps needed has been shown to be NP-complete, with a polynomial-time approximation algorithm for paths, and NP-complete for colored tokens, even with as few as two colors [164]. Various results have been shown for a "subset" variant, in which each token has a set of possible destinations [157].…”

“…Determining the shortest transformation sequence, TOKEN SWAPPING, is NP-complete [156], even on graphs of treewidth at most two [157], W[1]-hard with respect to the length of the reconfiguration sequence [157], and inapproximable [156]. Algorithmic results include polynomial-time solutions on paths [82], cycles [82], complete graphs [158], stars [159], complete bipartite graphs [139], and complete split graphs [160], as well as approximation algorithms for squares of paths [161], trees [139,156], and general graphs [156], fixed-parameter algorithms for nowhere dense graphs, which include planar graphs and graphs of bounded treewidth [157], and exact algorithms (with matching lower bounds) [156].…”

Introduction to Reconfiguration

“…In "parallel" variants, swapping is allowed on non-adjacent edges at the same time; the problem of determining the minimum number of swaps needed has been shown to be NP-complete, with a polynomial-time approximation algorithm for paths, and NP-complete for colored tokens, even with as few as two colors [163]. Various results have been shown for a "subset" variant, in which each token has a set of possible destinations [156].…”

“…Determining the shortest transformation sequence, TOKEN SWAPPING, is NP-complete [155], even on graphs of treewidth at most two [156], W[1]-hard with respect to the length of the reconfiguration sequence [156], and inapproximable [155]. Algorithmic results include polynomial-time solutions on paths [82], cycles [82], complete graphs [157], stars [158], complete bipartite graphs [138], and complete split graphs [159], as well as approximation algorithms for squares of paths [160], trees [138,155], and general graphs [155], fixed-parameter algorithms for nowhere dense graphs, which include planar graphs and graphs of bounded treewidth [156], and exact algorithms (with matching lower bounds) [155].…”

Finding Optimal Solutions to Token Swapping by Conflict-Based Search and Reduction to SAT

“…The TSWAP problem has been introduced recently. So far theoretical results concerning computational complexity [6] and approximations [7], [8] have appeared. To our best knowledge there is no study dealing with practical solving of TSWAP optimally.…”