Swapping Labeled Tokens on Graphs

Abstract

Abstract. Consider a puzzle consisting of n tokens on an n-vertex graph, where each token has a distinct starting vertex and a distinct target vertex it wants to reach, and the only allowed transformation is to swap the tokens on adjacent vertices. We prove that every such puzzle is solvable in O(n 2 ) token swaps, and thus focus on the problem of minimizing the number of token swaps to reach the target token placement. We give a polynomial-time 2-approximation algorithm for trees, and using this, obtain a polynomial-time 2α-approximation algorithm for graphs whose tree α-spanners can be computed in polynomial time. Finally, we show that the problem can be solved exactly in polynomial time on complete bipartite graphs.

Geometry and Generation of a New Graph Planarity Game

“…The next lemma shows that quadratically many swaps are sufficient; the result has been proved before as node swapping [20].…”

Introduction to Reconfiguration

“…In the years since it was first considered, curiosity about the problem has resulted in generalizing the problem, starting with problems using n 2 − 1 tiles arranged in a n × n square and moving to arbitrary arrangements represented as graphs. The problem has since been generalized further and has been applied to various scenarios, such as sorting [138] and robot motion [139,140]. Other variants have considered different ways of defining reconfiguration steps, including the sliding of a token any distance along a path unoccupied by tokens [141], the movement of a token along a path by a sequence of swaps [142], and the sliding and rotation of squares [143], as well as the consideration of directed graphs [140].…”

“…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].…”

The Connectivity of Token Graphs

“…From this new way of defining F k (G) is not hard to see that the study of the structure of F k (G) is intimately related to the class of problems known as reconfiguration problems. For more on reconfiguration problems, see for example [5,7,13,18,14,22,23]. On the other hand, most of the results reported in [11] address the following classical approach: Given a graph G and a graph invariant η; What can we say about η(F k (G)), when η(G) is known?…”