关于围长与Token滑动和Token跳跃问题的参数化复杂度
On girth and the parameterized complexity of token sliding and token jumping
AI总结：
该研究探讨Token滑动与Token跳跃问题的参数化复杂度，分析无环图类的影响，得出两类问题在特定无环图上固定参数可解、存在多项式核，同时在更长环排除的图类上W[1]难的结论。
摘要：
在Token Jumping（令牌跳跃）问题中，给定图$G = (V,E)$以及$G$的两个大小均为$k \geq 1$的独立集$S$和$T$。目标是判断$G$中是否存在一个大小为$k$的独立集序列$\langle S_0, S_1, \ldots, S_\ell \rangle$，使得对每个$i$，都满足$|S_i| = k$、$S_i$是独立集、$S = S_0$、$S_\ell = T$，且$|S_i ΔS_{i+1}| = 2$。换言之，若将每个独立集视为放置在$G$的顶点子集上的一组令牌，该问题就是要寻找一个独立集序列，通过单个令牌的跳跃操作将$S$转换为$T$，且全程保持集合的独立性。已知该问题在受限性极强的图类上是PSPACE完全的，例如平面有界度图和有界带宽图。\n\n与该问题密切相关的是Token Sliding（令牌滑动）问题：该问题不允许令牌跳跃到图的任意顶点，而是要求令牌沿着图的边滑动。Token Sliding问题在上述图类上也被证明是PSPACE完全的。我们研究了这两个问题在若干图类上的参数化复杂度，重点关注从输入图中排除特定环所带来的影响。具体而言，我们证明了当以$k$为参数时，Token Sliding和Token Jumping在无$C_4$二分图上都是固定参数可解的。对于Token Jumping，我们进一步证明该问题在无$\{C_3,C_4\}$图上存在多项式核。对于Token Sliding，我们同样证明该问题在有界度二分图上存在多项式核，我们认为这两项结果本身都具有独立研究价值。\n\n为补充这些正面结果，我们还证明：对于任意常数$p \geq 4$，这两个问题在无$\{C_4, \dots, C_p\}$图上都是W[1]难的，且Token Sliding即使在二分图上仍保持W[1]难的性质。
英文摘要：
In the Token Jumping problem we are given a graph $G = (V,E)$ and two independent sets $S$ and $T$ of $G$, each of size $k \geq 1$. The goal is to determine whether there exists a sequence of $k$-sized independent sets in $G$, $\langle S_0, S_1, \ldots, S_\ell \rangle$, such that for every $i$, $|S_i| = k$, $S_i$ is an independent set, $S = S_0$, $S_\ell = T$, and $|S_i ΔS_{i+1}| = 2$. In other words, if we view each independent set as a collection of tokens placed on a subset of the vertices of $G$, then the problem asks for a sequence of independent sets which transforms $S$ to $T$ by individual token jumps which maintain the independence of the sets. This problem is known to be PSPACE-complete on very restricted graph classes, e.g., planar bounded degree graphs and graphs of bounded bandwidth. A closely related problem is the Token Sliding problem, where instead of allowing a token to jump to any vertex of the graph we instead require that a token slides along an edge of the graph. Token Sliding is also known to be PSPACE-complete on the aforementioned graph classes. We investigate the parameterized complexity of both problems on several graph classes, focusing on the effect of excluding certain cycles from the input graph. In particular, we show that both Token Sliding and Token Jumping are fixed-parameter tractable on $C_4$-free bipartite graphs when parameterized by $k$. For Token Jumping, we in fact show that the problem admits a polynomial kernel on $\{C_3,C_4\}$-free graphs. In the case of Token Sliding, we also show that the problem admits a polynomial kernel on bipartite graphs of bounded degree. We believe both of these results to be of independent interest. We complement these positive results by showing that, for any constant $p \geq 4$, both problems are W[1]-hard on $\{C_4, \dots, C_p\}$-free graphs and Token Sliding remains W[1]-hard even on bipartite graphs.
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