SIAM Journal on Computing, Volume 49, Issue 6, Page 1397-1422, January 2020.
Bidimensionality theory was introduced by [E. D. Demaine et al., J. ACM, 52 (2005), pp. 866--893] as a tool to obtain subexponential time parameterized algorithms on H-minor-free graphs. In [E. D. Demaine and M. Hajiaghayi, Bidimensionality: New connections between FPT algorithms and PTASs, in Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, Philadelphia, 2005, pp. 590--601] this theory was extended in order to obtain polynomial time approximation schemes (PTASs) for bidimensional problems. In this work, we establish a third meta-algorithmic direction for bidimensionality theory by relating it to the existence of linear kernels for parameterized problems. In particular, we prove that every minor (resp., contraction) bidimensional problem that satisfies a separation property and is expressible in Countable Monadic Second Order Logic (CMSO) admits a linear kernel for classes of graphs that exclude a fixed graph (resp., an apex graph) H as a minor. Our results imply that a multitude of bidimensional problems admit linear kernels on the corresponding graph classes. For most of these problems no polynomial kernels on H-minor-free graphs were known prior to our work.

中文翻译：

---

二维与核

SIAM计算杂志，第49卷，第6期，第1397-1422页，2020年1月。
二维性理论是由[ED Demaine等人，J。ACM，52（2005），第866--893页]引入的，它是一种在无H小图上获得次指数时间参数化算法的工具。在[ED Demaine和M. Hajiaghayi，《二维性：FPT算法与PTAS之间的新联系》，在第16届ACM-SIAM离散算法研讨会（SODA）的会议记录中，SIAM，费城，2005年，第590--601页]对理论进行了扩展，以获得二维问题的多项式时间逼近方案（PTAS）。在这项工作中，我们通过将二维理论与参数化问题的线性核的存在联系起来，为二维理论建立了第三个元算法方向。特别是，我们证明每个未成年人（分别是 满足分离性质并在可数单子二阶逻辑（CMSO）中表示的二维问题允许线性核用于图的一类图，该图的核心排除了次要的固定图（例如顶点图）H。我们的结果表明，大量的二维问题允许线性核出现在相应的图类上。对于这些问题中的大多数，在我们进行工作之前，尚无H次要图上的多项式核。