Twin-width and polynomial kernels

TL;DR

This paper explores the limits of polynomial kernels for certain problems on graphs with bounded twin-width, establishing lower bounds for some problems and providing polynomial kernels for others, along with efficient algorithms for graphs of twin-width 1.

Contribution

It proves that polynomial kernels for k-Dominating Set on graphs with twin-width at most 4 are unlikely, and offers new polynomial kernels for Connected and Capacitated k-Vertex Cover.

Findings

Polynomial kernel for k-Dominating Set unlikely on graphs with twin-width ≤ 4.

Quadratic vertex kernel for Connected k-Vertex Cover on bounded twin-width graphs.

Deciding twin-width ≤ 1 can be done in polynomial time, enabling efficient solutions for many problems.

Abstract

We study the existence of polynomial kernels, for parameterized problems without a polynomial kernel on general graphs, when restricted to graphs of bounded twin-width. Our main result is that a polynomial kernel for $k$-Dominating Set on graphs of twin-width at most 4 would contradict a standard complexity-theoretic assumption. The reduction is quite involved, especially to get the twin-width upper bound down to 4, and can be tweaked to work for Connected $k$-Dominating Set and Total $k$-Dominating Set (albeit with a worse upper bound on the twin-width). The $k$-Independent Set problem admits the same lower bound by a much simpler argument, previously observed [ICALP '21], which extends to $k$-Independent Dominating Set, $k$-Path, $k$-Induced Path, $k$-Induced Matching, etc. On the positive side, we obtain a simple quadratic vertex kernel for Connected $k$-Vertex Cover and Capacitated $k$-Vertex Cover on graphs of bounded twin-width. Interestingly the kernel applies to graphs of Vapnik-Chervonenkis density 1, and does not require a witness sequence. We also present a more intricate $O(k^{1.5})$ vertex kernel for Connected $k$-Vertex Cover. Finally we show that deciding if a graph has twin-width at most 1 can be done in polynomial time, and observe that most optimization/decision graph problems can be solved in polynomial time on graphs of twin-width at most 1.