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

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

Transcription factor c-Myb inhibits breast cancer lung metastasis by suppression of tumor cell seeding

Metastasis accounts for most of cancer-related deaths. Paracrine signaling between tumor cells and the stroma induces changes in the tumor microenvironment required for metastasis. Transcription factor c-Myb was associated with breast cancer (BC) progression but its role in metastasis remains unclear. Here we show that increased c-Myb expression in BC cells inhibits spontaneous lung metastasis through impaired tumor cell extravasation. On contrary, BC cells with increased lung metastatic capacity exhibited low c-Myb levels. We identified a specific inflammatory signature, including Ccl2 chemokine, that was expressed in lung metastatic cells but was suppressed in tumor cells with higher c-Myb levels. Tumor cell-derived Ccl2 expression facilitated lung metastasis and rescued trans-endothelial migration of c-Myb overexpressing cells. Clinical data show that the identified inflammatory signature, together with a MYB expression, predicts lung metastasis relapse in BC patients. These results demonstrate that the c-Myb-regulated transcriptional program in BCs results in a blunted inflammatory response and consequently suppresses lung metastasis.Oncogene advance online publication, 30 October 2017; doi:10.1038/onc.2017.392