Faster Parameterized Vertex Multicut

TL;DR

The main technical contribution is a refined shadow removal step for vertex separation problems that only introduces an overhead of $k^{O(k)}\log n$ time.

Abstract

In the {\sc Vertex Multicut} problem the input consists of a graph $G$, integer $k$, and a set $\mathbf{T} = \{(s_1, t_1), \ldots, (s_p, t_p)\}$ of pairs of vertices of $G$. The task is to find a set $X$ of at most $k$ vertices such that, for every $(s_i, t_i) \in \mathbf{T}$, there is no path from $s_i$ to $t_i$ in $G - X$. Marx and Razgon [STOC 2011 and SICOMP 2014] and Bousquet, Daligault, and Thomassé [STOC 2011 and SICOMP 2018] independently and simultaneously gave the first algorithms for {\sc Vertex Multicut} with running time $f(k)n^{O(1)}$. The running time of their algorithms is $2^{O(k^3)}n^{O(1)}$ and $2^{O(k^{O(1)})}n^{O(1)}$, respectively. As part of their result, Marx and Razgon introduce the {\em shadow removal} technique, which was subsequently applied in algorithms for several parameterized cut and separation problems. The shadow removal step is the only step of the algorithm of Marx and Razgon which requires $2^{O(k^3)}n^{O(1)}$ time. Chitnis et al. [TALG 2015] gave an improved version of the shadow removal step, which, among other results, led to a $k^{O(k^2)}n^{O(1)}$ time algorithm for {\sc Vertex Multicut}. We give a faster algorithm for the {\sc Vertex Multicut} problem with running time $k^{O(k)}n^{O(1)}$. Our main technical contribution is a refined shadow removal step for vertex separation problems that only introduces an overhead of $k^{O(k)}\log n$ time. The new shadow removal step implies a $k^{O(k^2)}n^{O(1)}$ time algorithm for {\sc Directed Subset Feedback Vertex Set} and a $k^{O(k)}n^{O(1)}$ time algorithm for {\sc Directed Multiway Cut}, improving over the previously best known algorithms of Chitnis et al. [TALG 2015].

Figures (3)

• Figure 1: Overall framework of the algorithm for Vertex Multicut.
• Figure 2: An example for regions, where $W=\{w_1,w_2,w_3\}$. The diagram shows how the $2^3=8$ regions $P_{W}(G,R)$ (one for each $R\subseteq W$) are formed by the intersections of $\mathrm{FC}_{G,W}(w_1)$, $\mathrm{FC}_{G,W}(w_2)$, and $\mathrm{FC}_{G,W}(w_3)$ (and their complements). Together, these $8$ disjoint regions form a complete partition of the entire vertex set.
• Figure 3: An illustration for the contraction in the proof of Lemma \ref{['lem:bipedal']}, where $W = \{w_1,w_2,w_3\}$. In this contraction, the vertices in $B(P_{W}({\{w_i\}}))$ are contracted into $w_i$ for each $w_i\in W$.

Theorems & Definitions (76)

• Theorem 1: Main
• Lemma 1
• Definition 1: Important Separator
• Lemma 2
• Lemma 3
• proof
• Lemma 4
• Definition 2: Vertex Multicut
• Definition 3: Vertex Multiway Cut
• Lemma 5