Token Swapping on Trees

03/16/2019
by   Ahmad Biniaz, et al.
0

The input to the token swapping problem is a graph with vertices v_1, v_2, ..., v_n, and n tokens with labels 1, 2, ..., n, one on each vertex. The goal is to get token i to vertex v_i for all i= 1, ..., n using a minimum number of swaps, where a swap exchanges the tokens on the endpoints of an edge. Token swapping on a tree, also known as "sorting with a transposition tree", is not known to be in P nor NP-complete. We present some partial results: 1. An optimum swap sequence may need to perform a swap on a leaf vertex that has the correct token (a "happy leaf"), disproving a conjecture of Vaughan. 2. Any algorithm that fixes happy leaves---as all known approximation algorithms for the problem do---has approximation factor at least 4/3. Furthermore, the two best-known 2-approximation algorithms have approximation factor exactly 2. 3. A generalized problem---weighted coloured token swapping---is NP-complete on trees, but solvable in polynomial time on paths and stars. In this version, tokens and vertices have colours, and colours have weights. The goal is to get every token to a vertex of the same colour, and the cost of a swap is the sum of the weights of the two tokens involved.

READ FULL TEXT
research
03/11/2021

Hardness of Token Swapping on Trees

Given a graph where every vertex has exactly one labeled token, how can ...
research
03/19/2018

Swapping Colored Tokens on Graphs

We investigate the computational complexity of the following problem. We...
research
03/22/2022

TS-Reconfiguration of k-Path Vertex Covers in Caterpillars

A k-path vertex cover (k-PVC) of a graph G is a vertex subset I such tha...
research
08/19/2018

On Design of Problem Token Questions in Quality of Experience Surveys

User surveys for Quality of Experience (QoE) are a critical source of in...
research
06/21/2018

Shortest Reconfiguration Sequence for Sliding Tokens on Spiders

Suppose that two independent sets I and J of a graph with I = J are ...
research
06/25/2018

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

We study practical approaches to solving the token swapping (TSWAP) prob...
research
03/26/2017

Token-based Function Computation with Memory

In distributed function computation, each node has an initial value and ...

Please sign up or login with your details

Forgot password? Click here to reset