On Weighted Graph Separation Problems and Flow-Augmentation

One of the first application of the recently introduced technique of\emph{flow-augmentation} [Kim et al., STOC 2022] is a fixed-parameter algorithmfor the weighted version of \textsc{Directed Feedback Vertex Set}, a landmarkproblem in parameterized complexity. In this note we explore applicability offlow-augmentation to other weighted graph separation problems parameterized bythe size of the cutset. We show the following. -- In weighted undirected graphs\textsc{Multicut} is FPT, both in the edge- and vertex-deletion version. -- Theweighted version of \textsc{Group Feedback Vertex Set} is FPT, even with anoracle access to group operations. -- The weighted version of \textsc{DirectedSubset Feedback Vertex Set} is FPT. Our study reveals \textsc{DirectedSymmetric Multicut} as the next important graph separation problem whoseparameterized complexity remains unknown, even in the unweighted setting.<br

Taming Graphs with No Large Creatures and Skinny Ladders

We confirm a conjecture of Gartland and Lokshtanov [arXiv:2007.08761]: if for a hereditary graph class ? there exists a constant k such that no member of ? contains a k-creature as an induced subgraph or a k-skinny-ladder as an induced minor, then there exists a polynomial p such that every G ? ? contains at most p(|V(G)|) minimal separators. By a result of Fomin, Todinca, and Villanger [SIAM J. Comput. 2015] the latter entails the existence of polynomial-time algorithms for Maximum Weight Independent Set, Feedback Vertex Set and many other problems, when restricted to an input graph from ?. Furthermore, as shown by Gartland and Lokshtanov, our result implies a full dichotomy of hereditary graph classes defined by a finite set of forbidden induced subgraphs into tame (admitting a polynomial bound of the number of minimal separators) and feral (containing infinitely many graphs with exponential number of minimal separators)

Composition of bioactive secondary metabolites and mutagenicity of Sambucus nigra L. Fruit at different stages of ripeness

The relationship between the content of bioactive compounds and mutagenic activity of elderberry fruit at different stages of ripeness was investigated. Significant differences in the antioxidant profiles (TLC, HPLC with post-column derivatization) and antioxidant activity (ABTS, DPPH, and FC tests) were observed for studied elderberry extracts. The more ripen the fruit at the time of harvest were, the higher the content of anthocyanins (increase from 0 to 7.8 mg g−1d.w.) and antioxidant activity of the extracts (about 5-fold increase) were. Cyanogenic glycosides were not detected at any stage of ripeness. Accordingly, Ames MPF test (Xenometrix) did not reveal any mutagenicity. Our study suggests that instability of cyanogenic glycosides ensures safety of food/pharmaceutical products based on even not fully ripen elderberry fruit

A tight quasi-polynomial bound for Global Label Min-Cut

We study a generalization of the classic Global Min-Cut problem, called Global Label Min-Cut (or sometimes Global Hedge Min-Cut): the edges of the input (multi)graph are labeled (or partitioned into color classes or hedges), and removing all edges of the same label (color or from the same hedge) costs one. The problem asks to disconnect the graph at minimum cost. While the $st$-cut version of the problem is known to be NP-hard, the above global cut version is known to admit a quasi-polynomial randomized $n^{O(\log \mathrm{OPT})}$-time algorithm due to Ghaffari, Karger, and Panigrahi [SODA 2017]. They consider this as ``strong evidence that this problem is in P''. We show that this is actually not the case. We complete the study of the complexity of the Global Label Min-Cut problem by showing that the quasi-polynomial running time is probably optimal: We show that the existence of an algorithm with running time $(np)^{o(\log n/ (\log \log n)^2)}$ would contradict the Exponential Time Hypothesis, where $n$ is the number of vertices, and $p$ is the number of labels in the input. The key step for the lower bound is a proof that Global Label Min-Cut is W[1]-hard when parameterized by the number of uncut labels. In other words, the problem is difficult in the regime where almost all labels need to be cut to disconnect the graph. To turn this lower bound into a quasi-polynomial-time lower bound, we also needed to revisit the framework due to Marx [Theory Comput. 2010] of proving lower bounds assuming Exponential Time Hypothesis through the Subgraph Isomorphism problem parameterized by the number of edges of the pattern. Here, we provide an alternative simplified proof of the hardness of this problem that is more versatile with respect to the choice of the regimes of the parameters

Fixed-Parameter Tractability of Directed Multicut with Three Terminal Pairs Parameterized by the Size of the Cutset: Twin-width Meets Flow-Augmentation

We show fixed-parameter tractability of the Directed Multicut problem withthree terminal pairs (with a randomized algorithm). This problem, given adirected graph $G$, pairs of vertices (called terminals) $(s_1,t_1)$,$(s_2,t_2)$, and $(s_3,t_3)$, and an integer $k$, asks to find a set of at most$k$ non-terminal vertices in $G$ that intersect all $s_1t_1$-paths, all$s_2t_2$-paths, and all $s_3t_3$-paths. The parameterized complexity of thiscase has been open since Chitnis, Cygan, Hajiaghayi, and Marx provedfixed-parameter tractability of the 2-terminal-pairs case at SODA 2012, andPilipczuk and Wahlstr\"{o}m proved the W[1]-hardness of the 4-terminal-pairscase at SODA 2016. On the technical side, we use two recent developments in parameterizedalgorithms. Using the technique of directed flow-augmentation [Kim, Kratsch,Pilipczuk, Wahlstr\"{o}m, STOC 2022] we cast the problem as a CSP problem withfew variables and constraints over a large ordered domain.We observe that thisproblem can be in turn encoded as an FO model-checking task over a structureconsisting of a few 0-1 matrices. We look at this problem through the lenses oftwin-width, a recently introduced structural parameter [Bonnet, Kim,Thomass\'{e}, Watrigant, FOCS 2020]: By a recent characterization [Bonnet,Giocanti, Ossona de Mendes, Simon, Thomass\'{e}, Toru\'{n}czyk, STOC 2022] thesaid FO model-checking task can be done in FPT time if the said matrices havebounded grid rank. To complete the proof, we show an irrelevant vertex rule: Ifany of the matrices in the said encoding has a large grid minor, a vertexcorresponding to the ``middle'' box in the grid minor can be proclaimedirrelevant -- not contained in the sought solution -- and thus reduced.<br

Fixed-parameter tractability of Directed Multicut with three terminal pairs parameterized by the size of the cutset: twin-width meets flow-augmentation

We show fixed-parameter tractability of the Directed Multicut problem with three terminal pairs (with a randomized algorithm). This problem, given a directed graph $G$, pairs of vertices (called terminals) $(s_1,t_1)$, $(s_2,t_2)$, and $(s_3,t_3)$, and an integer $k$, asks to find a set of at most $k$ non-terminal vertices in $G$ that intersect all $s_1t_1$-paths, all $s_2t_2$-paths, and all $s_3t_3$-paths. The parameterized complexity of this case has been open since Chitnis, Cygan, Hajiaghayi, and Marx proved fixed-parameter tractability of the 2-terminal-pairs case at SODA 2012, and Pilipczuk and Wahlstr\"{o}m proved the W[1]-hardness of the 4-terminal-pairs case at SODA 2016. On the technical side, we use two recent developments in parameterized algorithms. Using the technique of directed flow-augmentation [Kim, Kratsch, Pilipczuk, Wahlstr\"{o}m, STOC 2022] we cast the problem as a CSP problem with few variables and constraints over a large ordered domain.We observe that this problem can be in turn encoded as an FO model-checking task over a structure consisting of a few 0-1 matrices. We look at this problem through the lenses of twin-width, a recently introduced structural parameter [Bonnet, Kim, Thomass\'{e}, Watrigant, FOCS 2020]: By a recent characterization [Bonnet, Giocanti, Ossona de Mendes, Simon, Thomass\'{e}, Toru\'{n}czyk, STOC 2022] the said FO model-checking task can be done in FPT time if the said matrices have bounded grid rank. To complete the proof, we show an irrelevant vertex rule: If any of the matrices in the said encoding has a large grid minor, a vertex corresponding to the ``middle'' box in the grid minor can be proclaimed irrelevant -- not contained in the sought solution -- and thus reduced

Jones’ conjecture in subcubic graphs

We confirm Jones' Conjecture for subcubic graphs. Namely, if a subcubic planar graph does not contain k + 1 vertex-disjoint cycles, then it suffices to delete 2k vertices to obtain a forest