We present new min-max relations in digraphs between the number of paths satisfying certain conditions and the order of the corresponding cuts. We define these objects in order to capture, in the context of solving the half-integral linkage problem, the essential properties needed for reaching a large bramble of constant congestion from the terminal set. This strategy has been used ad-hoc in several articles, usually with lengthy technical proofs, and our objective is to abstract it to make it applicable in a simpler and unified way. We provide two proofs of the min-max relations, one consisting in applying Menger’s Theorem on appropriately defined digraphs, and an alternative simpler one using matroids, however with worse polynomial running time. As an application, we manage to simplify and improve several results of Edwards et al. in 2017 and of Giannopoulou et al. in 2022 about finding half-integral linkages in digraphs. Concerning the former, besides being simpler, our proof provides an almost optimal bound on the strong connectivity of a digraph for it to be half-integrally feasible under the presence of a large bramble of congestion two (or equivalently, if the directed tree-width is large). Concerning the latter, our proof uses brambles as rerouting objects instead of cylindrical grids, hence yielding much better bounds and being somehow independent of a particular topology. We hope that our min-max relations will find further applications as, in our opinion, they are simple, robust, and versatile to be easily applicable to different types of routing problems in digraphs.

International audience

The grid theorem of Robertson and Seymour [ J. Combin. Theory Ser. B, 41 (1986), pp. 92--114] is one of the most important tools in the field of structural graph theory, finding numerous applications in the design of algorithms for undirected graphs. An analogous version of the grid theorem in digraphs was conjectured by Johnson et al. [ J. Combin. Theory Ser. B, 82 (2001), pp. 138--154] and proved by Kawarabayashi and Kreutzer [ Proceedings of STOC, 2015, pp. 655--664]. Namely, they showed that there is a function $f(k)$ such that every digraph of directed tree-width at least $f(k)$ contains a cylindrical grid of order $k$ as a butterfly minor, and stated that their proof can be turned into an \sf XP algorithm, with parameter $k$, that either constructs a decomposition of the appropriate width or finds the claimed large cylindrical grid as a butterfly minor. In this paper, we adapt some of the steps of the proof of Kawarabayashi and Kreutzer to improve this \sf XP algorithm into a fixed-parameter tractable (\sf FPT) algorithm. Toward this, our main technical contributions are two \sf FPT algorithms with parameter $k$. The first one either produces an arboreal decomposition of width $3k-2$ or finds a haven of order $k$ in a digraph $D$, improving on the original result for arboreal decompositions by Johnson et al. [ J. Combin. Theory Ser. B, 82 (2001), pp. 138--154]. The second algorithm finds a well-linked set of order $k$ in a digraph $D$ of large directed tree-width. As tools to prove these results, we show how to solve a generalized version of the problem of finding balanced separators for a given set of vertices $T$ in \sf FPT time with parameter $|T|$, a result that we consider to be of its own interest.

In the Maximum Minimal Vertex Cover (MMVC) problem, we are given a graph G and a positive integer k, and the objective is to decide whether G contains a minimal vertex cover of size at least k. Motivated by the kernelization of MMVC with parameter k, our main contribution is to introduce a simple general framework to obtain lower bounds on the degrees of a certain type of polynomial kernels for vertex-optimization problems, which we call {lop-kernels}. Informally, this type of kernels is required to preserve large optimal solutions in the reduced instance, and captures the vast majority of existing kernels in the literature. As a consequence of this framework, we show that the trivial quadratic kernel for MMVC is essentially optimal, answering a question of Boria et al. [Discret. Appl. Math. 2015], and that the known cubic kernel for Maximum Minimal Feedback Vertex Set is also essentially optimal. On the positive side, given the (plausible) non-existence of subquadratic kernels for MMVC on general graphs, we provide subquadratic kernels on H-free graphs for several graphs H, such as the bull, the paw, or the complete graphs, by making use of the Erdős-Hajnal property in order to find an appropriate decomposition. Finally, we prove that MMVC does not admit polynomial kernels parameterized by the size of a minimum vertex cover of the input graph, even on bipartite graphs, unless NP ⊆ coNP / poly. This indicates that parameters smaller than the solution size are unlike to yield polynomial kernels for MMVC.

Given a graph G, a properk-coloring of G is a partition $$c = (S_i)_{i\in [1,k]}$$ of V(G) into k stable sets $$S_1,\ldots , S_{k}$$ . Given a weight function $$w: V(G) \rightarrow {\mathbb {R}}^+$$ , the weight of a color $$S_i$$ is defined as $$w(i) = \max _{v \in S_i} w(v)$$ and the weight of a coloringc as $$w(c) = \sum _{i=1}^{k}w(i)$$ . Guan and Zhu (Inf Process Lett 61(2):77–81, 1997) defined the weighted chromatic number of a pair (G, w), denoted by $$\sigma (G,w)$$ , as the minimum weight of a proper coloring of G. The problem of determining $$\sigma (G,w)$$ has received considerable attention during the last years, and has been proved to be notoriously hard: for instance, it is NP-hard on split graphs, unsolvable on n-vertex trees in time $$n^{o(\log n)}$$ unless the ETH fails, and W[1]-hard on forests parameterized by the size of a largest tree. We focus on the so-called dual parameterization of the problem: given a vertex-weighted graph (G, w) and an integer k, is $$\sigma (G,w) \le \sum _{v \in V(G)} w(v) - k$$ ? This parameterization has been recently considered by Escoffier (in: Proceedings of the 42nd international workshop on graph-theoretic concepts in computer science (WG). LNCS, vol 9941, pp 50–61, 2016), who provided an FPT algorithm running in time $$2^{{\mathcal {O}}(k \log k)} \cdot n^{{\mathcal {O}}(1)}$$ , and asked which kernel size can be achieved for the problem. We provide an FPT algorithm in time $$9^k \cdot n^{{\mathcal {O}}(1)}$$ , and prove that no algorithm in time $$2^{o(k)} \cdot n^{{\mathcal {O}}(1)}$$ exists under the ETH. On the other hand, we present a kernel with at most $$(2^{k-1}+1) (k-1)$$ vertices, and rule out the existence of polynomial kernels unless $$\mathsf{NP} \subseteq \mathsf{coNP} / \mathsf{poly}$$ , even on split graphs with only two different weights. Finally, we identify classes of graphs allowing for polynomial kernels, namely interval graphs, comparability graphs, and subclasses of circular-arc and split graphs, and in the latter case we present lower bounds on the degrees of the polynomials.

For two positive integers k and $$\ell $$ ℓ , a $$(k \times \ell )$$ ( k × ℓ ) -spindle is the union of k pairwise internally vertex-disjoint directed paths with $$\ell $$ ℓ arcs each between two vertices u and v. We are interested in the (parameterized) complexity of several problems consisting in deciding whether a given digraph contains a subdivision of a spindle, which generalize both the Maximum Flow and Longest Path problems. We obtain the following complexity dichotomy: for a fixed $$\ell \ge 1$$ ℓ ≥ 1 , finding the largest k such that an input digraph G contains a subdivision of a $$(k \times \ell )$$ ( k × ℓ ) -spindle is polynomial-time solvable if $$\ell \le 3$$ ℓ ≤ 3 , and NP-hard otherwise. We place special emphasis on finding spindles with exactly two paths and present FPT algorithms that are asymptotically optimal under the ETH. These algorithms are based on the technique of representative families in matroids, and use also color-coding as a subroutine. Finally, we study the case where the input graph is acyclic, and present several algorithmic and hardness results.

Given a graph $G$, a proper $k$-coloring of $G$ is a partition $c = (S_i)_{i\in [1,k]}$ of $V(G)$ into $k$ stable sets $S_1,\ldots, S_{k}$. Given a weight function $w: V(G) \to \mathbb{R}^+$, the weight of a color $S_i$ is defined as $w(i) = \max_{v \in S_i} w(v)$ and the weight of a coloring $c$ as $w(c) = \sum_{i=1}^{k}w(i)$. Guan and Zhu [Inf. Process. Lett., 1997] defined the weighted chromatic number of a pair $(G,w)$, denoted by $\sigma(G,w)$, as the minimum weight of a proper coloring of $G$. The problem of determining $\sigma(G,w)$ has received considerable attention during the last years, and has been proved to be notoriously hard: for instance, it is NP-hard on split graphs, unsolvable on $n$-vertex trees in time $n^{o(\log n)}$ unless the ETH fails, and W[1]-hard on forests parameterized by the size of a largest tree. In this article we provide some positive results for the problem, by considering its so-called dual parameterization: given a vertex-weighted graph $(G,w)$ and an integer $k$, the question is whether $\sigma(G,w) \leq \sum_{v \in V(G)} w(v) - k$. We prove that this problem is FPT by providing an algorithm running in time $9^k \cdot n^{O(1)}$, and it is easy to see that no algorithm in time $2^{o(k)} \cdot n^{O(1)}$ exists under the ETH. On the other hand, we present a kernel with at most $(2^{k-1}+1) (k-1)$ vertices, and we rule out the existence of polynomial kernels unless ${\sf NP} \subseteq {\sf coNP} / {\sf poly}$, even on split graphs with only two different weights. Finally, we identify some classes of graphs on which the problem admits a polynomial kernel, in particular interval graphs and subclasses of split graphs, and in the latter case we present lower bounds on the degrees of the polynomials.