Independent Set Discovery on Biclique-Free Graphs Is Fixed-Parameter Tractable

Abstract

\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.