We recently introduced the graph invariant twin-width, and showed that first-order model checking can be solved in time $f(d,k)n$ for $n$-vertex graphs given with a witness that the twin-width is at most $d$, called $d$-contraction sequence or $d$-sequence, and formulas of size $k$ [Bonnet et al., FOCS '20]. The inevitable price to pay for such a general result is that $f$ is a tower of exponentials of height roughly $k$. In this paper, we show that algorithms based on twin-width need not be impractical. We present $2^{O(k)}n$-time algorithms for $k$-Independent Set, $r$-Scattered Set, $k$-Clique, and $k$-Dominating Set when an $O(1)$-sequence is provided. We further show how to solve weighted $k$-Independent Set, Subgraph Isomorphism, and Induced Subgraph Isomorphism, in time $2^{O(k \log k)}n$. These algorithms are based on a dynamic programming scheme following the sequence of contractions forward. We then show a second algorithmic use of the contraction sequence, by starting at its end and rewinding it. As an example, we establish that bounded twin-width classes are $\chi$-bounded. This significantly extends the $\chi$-boundedness of bounded rank-width classes, and does so with a very concise proof. The third algorithmic use of twin-width builds on the second one. Playing the contraction sequence backward, we show that bounded twin-width graphs can be edge-partitioned into a linear number of bicliques, such that both sides of the bicliques are on consecutive vertices, in a fixed vertex ordering. Given that biclique edge-partition, we show how to solve the unweighted Single-Source Shortest Paths and hence All-Pairs Shortest Paths in sublinear time $O(n \log n)$ and time $O(n^2 \log n)$, respectively. Finally we show that Min Dominating Set and related problems have constant integrality gaps on bounded twin-width classes, thereby getting constant approximations on these classes.

翻译：我们最近引入了“硬度”双振度图, 并显示, 用于这种一般结果的不可避免价格是, 美元是高度的指数, 大约为美元。 在本文中, 我们显示基于双振度的“垂直”的算法并非不切实际。 我们在“ 双振” 上显示“ 美元”, 称为“ 美元”, 或“ 美元”, 以美元为序列, 以及以美元为单位的公式。 当提供了“ 美元” 二度“, FOS'20 ” 之后, 美元是不可避免的。 我们进一步展示了如何以美元为单位的“ 高度”, 以美元为单位的“ 方向”, 以“ 方向” 为单位的“ 方向”, 以“ 方向” 为单位的“ 方向”, 以“ 方向” 为单位的“ 方向”, 以“ 方向” 显示一个“ 方向” 。