Parameterized complexity has always been concerned with practical computing: by confining combinatorial explosion to a secondary parameter $k$, one can uncover why and how many NP-hard problems are effectively tackled in practice. Today, however, the scale of data has changed: scientists study Big Data, which is so large that even quadratic dependence in the total input size $n$ is unaffordable. Therefore, what constitutes a practical algorithm has also changed. Classically, parameterized complexity is blind to the difference between defining fixed parameter tractability multiplicatively (i.e. $f(k) \cdot n^c$) or additively (i.e. $f(k) + n^c$). But what if the constant $c$ is one and we require true linearity, is this distinction still inconsequential? Here, we define and explore Truly Linear FPT (TLFPT) -- that is $O(n)+f(k)$ -- and show that it is a strict subset of Linear FPT (LFPT) -- that is $O(n) \cdot f(k)$ -- via diagonalization. Populating TLFPT requires careful consideration of linear-time algorithmics and data structures. We meet many inhabitants of TLFPT: SAT, Vertex Cover, Min-Max Matching, $(n-k)$-Coloring, Diverse Pair of Matchings, $k$-Path, and $H$-Coloring. Our parameterizations are equally varied. Beyond classical parameters like solution size, we leverage two parameters, treedepth and BFS-width, which are particularly well-suited to the TLFPT regime. We do so by developing techniques based on depth- and breadth-first search. For parameterized complexity to be of service to the scientific community, we need to contend with Big Data. For sufficiently large inputs, FPT beyond linear may not suffice. Thus, there is a practical and theoretical need for more ambitious goals. TLFPT is a first step forward.

翻译：参数化复杂度始终关注实际计算：通过将组合爆炸限制在次要参数$k$中，可以揭示许多NP困难问题为何以及如何在实践中被有效解决。然而，如今数据规模已发生变化：科学家研究大数据，其规模之大使得即使总输入规模$n$的二次依赖关系也难以承受。因此，何为实用算法也已改变。经典上，参数化复杂度对固定参数可处理性以乘法形式（即$f(k) \cdot n^c$）还是加法形式（即$f(k) + n^c$）定义的区别视而不见。但若常数$c$为1且要求真正的线性关系，这种区分是否仍无关紧要？本文定义并探索了真正线性FPT（TLFPT）——即$O(n)+f(k)$——并通过对角线化方法证明其是线性FPT（LFPT）——即$O(n) \cdot f(k)$——的严格子集。构建TLFPT需要仔细考虑线性时间算法和数据结构。我们发现了TLFPT中的许多成员：SAT、顶点覆盖、最小-最大匹配、$(n-k)$染色、多样匹配对、$k$路径以及$H$染色。我们的参数化同样多样。除了解规模等经典参数外，我们利用了特别适合TLFPT框架的两个参数：树深度和BFS宽度。为此，我们通过基于深度优先搜索和广度优先搜索开发了相关技术。为了使参数化复杂度服务于科学界，我们需要应对大数据。对于足够大的输入，超越线性的FPT可能不足以解决问题。因此，从实践和理论角度都需要更宏大的目标。TLFPT是向前迈出的第一步。