
arXiv: 1903.06981
The input to the token swapping problem is a graph with vertices $v_1, v_2, \ldots, v_n$, and $n$ tokens with labels $1, 2, \ldots, n$, one on each vertex. The goal is to get token $i$ to vertex $v_i$ for all $i= 1, \ldots, n$ using a minimum number of swaps, where a swap exchanges the tokens on the endpoints of an edge. We present some results about token swapping on a tree, also known as "sorting with a transposition tree": 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, even when they are restricted to be subdivided stars, 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.
FOS: Computer and information sciences, 03D15, 05C05, 68R05, Analysis of algorithms and problem complexity, Computational Complexity (cs.CC), Cayley graph, 05C05, Graph algorithms (graph-theoretic aspects), 03d15, 05c05, 68r05, sorting with transposition tree, Computer Science - Data Structures and Algorithms, QA1-939, Data Structures and Algorithms (cs.DS), token swapping, f.2.0, cs.CC, reconfiguration, 68R05, computer science - data structures and algorithms, Computer Science - Computational Complexity, cs.DS, Graph theory (including graph drawing) in computer science, 03D15, Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.), computer science - computational complexity, F.2.0, Mathematics
FOS: Computer and information sciences, 03D15, 05C05, 68R05, Analysis of algorithms and problem complexity, Computational Complexity (cs.CC), Cayley graph, 05C05, Graph algorithms (graph-theoretic aspects), 03d15, 05c05, 68r05, sorting with transposition tree, Computer Science - Data Structures and Algorithms, QA1-939, Data Structures and Algorithms (cs.DS), token swapping, f.2.0, cs.CC, reconfiguration, 68R05, computer science - data structures and algorithms, Computer Science - Computational Complexity, cs.DS, Graph theory (including graph drawing) in computer science, 03D15, Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.), computer science - computational complexity, F.2.0, Mathematics
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