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.
翻译:象征性交换问题的输入是一张图, 上面写着 $_ 1, v_ 2, v_ 2, ldots, v_n$, 上面写着标签为 1, 2,\ldots, n$, 每个顶点上写着一个美元。 目标是让所有$i= 1,\ldots, n$的折叠美元到顶点的顶点 $exexexex $_ i$。 使用最低数量的折叠式, 交换边缘端端点上的标牌。 我们展示了树上的标牌交换的一些结果, 也称为“ 转换颜色树” 的符号交换结果 : 1. 最佳交换顺序可能需要在叶牌上换一个牌子( “ 快乐叶子” ), 扭曲Vauauvanan 的轮廓。 2. 任何固定快乐假期的算法 -- 所有已知的近端点算法都至少有 4/3 。 此外, 两种已知的2 Approd colog 运算算算法的值算算算算算算算算得至少2 的颜色值, 的颜色比值是 。 。 3 。 。 。 。 当质 质 质变数 值 目标 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值, 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 。 。 。