Twin-width and polynomial kernels

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-VertexCover 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.

Compact representation for matrices of bounded twin-width

“…The notion of twin-width was introduced very recently by Bonnet et al in [5], and has immediately gathered immense interest. As shown in [5] and in multiple subsequent works [1,2,3,4,6,10,11], twin-width is a versatile measure of complexity not only for matrices, but also for permutations and for graphs by considering a suitable matrix representation, which in the latter case is just the adjacency matrix. In particular, for every fixed t ∈ N, graphs excluding K t as a minor and graphs having cliquewidth at most t have bounded twin-width, which means that the concept of boundedness of twin-width is a vast generalization of boundedness of cliquewidth that does not assume tree-likeness of the structure of the graph.…”

“…In particular, for every fixed t ∈ N, graphs excluding K t as a minor and graphs having cliquewidth at most t have bounded twin-width, which means that the concept of boundedness of twin-width is a vast generalization of boundedness of cliquewidth that does not assume tree-likeness of the structure of the graph. As shown in the aforementioned works, this generalization is combinatorially rich [1,5,10], algorithmically useful [2,4,5], and exposes deep connections with notions studied in finite model theory [3,5,6,11]. In particular, assuming a suitable contraction sequence is provided on input, model-checking First-Order logic on graphs of bounded twin-width can be done in linear fixed-parameter time [5].…”

Bounds for the Twin-width of Graphs

“…It has been shown that every graph class of bounded twin-width is small 3 [4], χ-bounded [5], and admits a linear-time algorithm for first-order model checking [8] (if we are given a certificate for an input graph to have small twin-width). Despite being a relatively new concept, twin-width has already generated a large amount of interests [8,4,5,6,13,7,1,10,17,9].…”