[论文解读] Bounding the Mim-Width of Hereditary Graph Classes

Nick Brettell, Jake Horsfield+3 more|arXiv (Cornell University)|Jan 1, 2020

Advanced Graph Theory Research参考文献 12被引用 3

一句话总结

该论文证明了对于所有 s ≥ 0 和 t ≥ 1，(Kt, sP1 + P5)-free 图的 mim-宽度是有界的，且可在多项式时间内计算。通过利用大小为常数的支配集构造分支分解，并应用关于划分集合之间诱导匹配的拉姆齐理论界，作者证明了此类图对 k-Colouring 等 NP-难问题具有高效算法，从而为该类图中已知的可解性结果提供了统一解释。

ABSTRACT

A large number of NP-hard graph problems are solvable in XP time when parameterized by some width parameter. Hence, when solving problems on special graph classes, it is helpful to know if the graph class under consideration has bounded width. In this paper we consider mim-width, a particularly general width parameter that has a number of algorithmic applications whenever a decomposition is "quickly computable" for the graph class under consideration. We start by extending the toolkit for proving (un)boundedness of mim-width of graph classes. By combining our new techniques with known ones we then initiate a systematic study into bounding mim-width from the perspective of hereditary graph classes, and make a comparison with clique-width, a more restrictive width parameter that has been well studied. We prove that for a given graph H, the class of H-free graphs has bounded mim-width if and only if it has bounded clique-width. We show that the same is not true for (H₁,H₂)-free graphs. We identify several general classes of (H₁,H₂)-free graphs having unbounded clique-width, but bounded mim-width, illustrating the power of mim-width. Moreover, we show that a branch decomposition of constant mim-width can be found in polynomial time, for these classes. Hence, as mentioned, these results have algorithmic implications: when the input is restricted to such a class of (H₁,H₂)-free graphs, many problems become polynomial-time solvable, including classical problems such as k-Colouring and Independent Set, domination-type problems known as LC-VSVP problems, and distance versions of LC-VSVP problems, to name just a few. We also prove a number of new results showing that, for certain H₁ and H₂, the class of (H₁,H₂)-free graphs has unbounded mim-width. Boundedness of clique-width implies boundedness of mim-width. By combining our results, which give both new bounded and unbounded cases for mim-width, with the known bounded cases for clique-width, we present summary theorems of the current state of the art for the boundedness of mim-width for (H₁,H₂)-free graphs. In particular, we classify the mim-width of (H₁,H₂)-free graphs for all pairs (H₁,H₂) with |V(H₁)| + |V(H₂)| ≤ 8. When H₁ and H₂ are connected graphs, we classify all pairs (H₁,H₂) except for one remaining infinite family and a few isolated cases.

研究动机与目标

• 确定对于所有 s ≥ 0 和 t ≥ 1，(Kt, sP1 + P5)-free 图的 mim-宽度是否是有界的，并且可快速计算。
• 通过将 k-Colouring 在 (sP1 + P5)-free 图中的多项式时间可解性与有界 mim-宽度联系起来，为该类图中已知的可解性结果提供结构解释。
• 通过解决无穷多个开放情况，扩展对可遗传图类中 mim-宽度的理解。

提出的方法

• 通过关于禁止团 Kt 大小 t 的归纳法，证明 (Kt, sP1 + P5)-free 图的 mim-宽度有界。
• 在任意此类图中识别出一个大小为常数的支配集 D，利用 P5-自由性或 sP1 + P5-自由性，将 |D| 限制在 max{3, t−1} 或 s+4 以内。
• 根据与 D 中顶点的邻接关系，将顶点集 V ∓ D 划分为 p = |D| 个集合 X1, ..., Xp，确保每个 Xi 诱导出一个 (Kt−1, sP1 + P5)-free 子图。
• 应用拉姆齐定理，对任意 Xi 和 Xj (i ≠ j) 之间的诱导匹配大小进行界控，证明 cutmimG(Xi, Xj) < R(t−1, R(t−1, s+2))。
• 通过在 Xi 分量上递归构造，为 G − D 构造一个 mim-宽度有界的分支分解。
• 通过附加一个具有 |D|+2 片叶子的子三叉树，将分解扩展至包含 D，保持 mim-宽度在常数加法因子内。

实验结果

研究问题

• RQ1 对于所有 s ≥ 0 和 t ≥ 1，(Kt, sP1 + P5)-free 图的 mim-宽度是否是有界的，并且可在多项式时间内计算？
• RQ2 能否通过有界 mim-宽度解释 (sP1 + P5)-free 图中 k-Colouring 的多项式时间可解性？
• RQ3 (Kt, sP1 + P5)-free 图的哪些结构特性使得能够构造出低 mim-宽度的分支分解？

主要发现

• (Kt, sP1 + P5)-free 圖的 mim-寬度由僅依賴於 s 和 t 的常數界控，且此界可在多項式時間內計算。
• 對於每組固定的 s ≥ 0 和 t ≥ 1，可在多項式時間內為 (Kt, sP1 + P5)-free 圖計算出 mim-寬度為常數的分支分解。
• 證明確立了 (sP1 + P5)-free 圖類具有有界 mim-寬度，從而解釋了該類圖中 k-Colouring 的多項式時間可解性。
• 分解中任意兩部分 Xi 和 Xj 之間的最大诱导匹配大小受 R(t−1, R(t−1, s+2)) 界控，確保 mim-寬度有界。
• 分支分解的構造是高效的，依賴於大小為常數的支配集以及對誘導子圖的遞歸分解。
• 該結果意味著所有在有界 mim-寬度圖上可解的 NP-難問題（例如 k-Colouring）在 (sP1 + P5)-free 圖上也具有多項式時間算法，其根本原因在於 mim-寬度有界。