双团自由图上的独立集发现问题是固定参数可处理的
Independent Set Discovery on Biclique-Free Graphs Is Fixed-Parameter Tractable
- Institute of Software, Chinese Academy of Sciences（中国科学院软件研究所）
- University of Regensburg（雷根斯堡大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该论文解决了双团自由图上独立集发现问题的开放复杂性，证明其在令牌数k上固定参数可处理，并给出确定性算法计算最小滑动次数及最优目标序列。
AI中文摘要：
\textnormal{\textsc{独立集发现}}问题询问：在互异顶点上的$k$个令牌的配置，是否可以通过一系列令牌滑动（每次将一个令牌移动到未占用的邻居）转换为一个独立的$k$-集；仅要求最终配置是独立的。\textnormal{\textsc{独立集发现}}是解发现中的核心问题：其目标不是预先给定的，必须与到达目标所需的令牌移动一起选择。Fellows等人证明了该问题在$k$上对每个固定的有界退化类和每个无处稠密类是FPT的，但留下了双团自由情形未解决。双团自由设置实质上超出了这两种情形：双团自由类可以具有无界退化，甚至可以是某处稠密的。我们肯定地解决了这个开放问题。给定一个$n$顶点、$m$边的图，并保证是$K_{d,d}$-自由的，以及一个初始的$k$令牌配置，我们的确定性算法计算最小滑动次数，并在时间$2^{O(dk\log k)}(n+m)^{O(1)}$内返回一个最优独立目标和一个最短的无碰撞滑动序列，或者证明没有独立目标可达。因此，该问题对每个固定的$d$在$k$上是FPT的，并且在$k+d$上是一致FPT的。证明结合了令牌移动的精确最小成本分配刻画与对候选列表的有界、成本相关的\emph{廉价前缀}的局部分支。该方法还产生了在具有重叠候选集的$K_{d,d}$-自由图上的加权独立横截的精确FPT算法，对有界退化图上两个问题的直接精确FPT算法，一个边数敏感的XP算法，对有界$s$-余度和不平衡双团排除的更紧界，以及在一个单独的定向图上加权移动的精确扩展。
英文摘要：
\textnormal{\textsc{Independent Set Discovery}} asks whether a configuration of $k$ tokens on distinct vertices can be transformed into an independent $k$-set by a sequence of token slides, each moving one token to an unoccupied neighbor; only the terminal configuration must be independent. \textnormal{\textsc{Independent Set Discovery}} is a central problem in solution discovery: its target is not prescribed and must be chosen together with the token movements needed to reach it. Fellows et al. proved it FPT in $k$ on every fixed bounded-degeneracy class and every nowhere-dense class, leaving the biclique-free case open. The biclique-free setting lies substantially beyond both regimes: biclique-free classes can have unbounded degeneracy and even be somewhere dense. We resolve the open problem affirmatively. Given an $n$-vertex, $m$-edge graph promised to be $K_{d,d}$-free and an initial $k$-token configuration, our deterministic algorithm computes the minimum number of slides and, in time $2^{O(dk\log k)}(n+m)^{O(1)}$, returns an optimal independent target and a shortest collision-free slide sequence or certifies that no independent target is reachable. Thus the problem is FPT in $k$ for every fixed $d$ and uniformly FPT in $k+d$. The proof combines an exact minimum-cost assignment characterization of token movement with local branching on bounded, cost-relevant \emph{cheap prefixes} of candidate lists. The method also yields an exact FPT algorithm for weighted independent transversals on $K_{d,d}$-free graphs with overlapping candidate sets, direct exact FPT algorithms for both problems on bounded-degeneracy graphs, an edge-count-sensitive XP algorithm, sharper bounds for bounded $s$-codegree and unbalanced biclique exclusion, and an exact extension to weighted movement on a separate directed graph.
