11 research outputs found
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
Colouring (Pr + Ps)-Free Graphs
The k-Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for a fixed integer k such that no two adjacent vertices are coloured alike. If each vertex u must be assigned a colour from a prescribed list L(u)⊆{1,…,k}, then we obtain the List k-Colouring problem. A graph G is H-free if G does not contain H as an induced subgraph. We continue an extensive study into the complexity of these two problems for H-free graphs. The graph Pr+Ps is the disjoint union of the r-vertex path Pr and the s-vertex path Ps. We prove that List 3-Colouring is polynomial-time solvable for (P2+P5)-free graphs and for (P3+P4)-free graphs. Combining our results with known results yields complete complexity classifications of 3-Colouring and List 3-Colouring on H-free graphs for all graphs H up to seven vertices
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 -cut version of the problem is known to be NP-hard, the above
global cut version is known to admit a quasi-polynomial randomized -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 would
contradict the Exponential Time Hypothesis, where is the number of
vertices, and 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 , pairs of vertices (called terminals) ,, and , and an integer , asks to find a set of at most non-terminal vertices in that intersect all -paths, all-paths, and all -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 , pairs of vertices (called terminals) ,
, and , and an integer , asks to find a set of at most
non-terminal vertices in that intersect all -paths, all
-paths, and all -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
Colouring (Pr+Ps)-free graphs.
The k-Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for a fixed integer k such that no two adjacent vertices are coloured alike. If each vertex u must be assigned a colour from a prescribed list L(u) subseteq {1,...,k}, then we obtain the List k-Colouring problem. A graph G is H-free if G does not contain H as an induced subgraph. We continue an extensive study into the complexity of these two problems for H-free graphs. We prove that List 3-Colouring is polynomial-time solvable for (P_2+P_5)-free graphs and for (P_3+P_4)-free graphs. Combining our results with known results yields complete complexity classifications of 3-Colouring and List 3-Colouring on H-free graphs for all graphs H up to seven vertices. We also prove that 5-Colouring is NP-complete for (P_3+P_5)-free graphs
Clique-Width: Harnessing the Power of Atoms
Many NP-complete graph problems are polynomial-time solvable on graph classes of bounded clique-width. Several of these problems are polynomial-time solvable on a hereditary graph class G if they are so on the atoms (graphs with no clique cut-set) of G . Hence, we initiate a systematic study into boundedness of clique-width of atoms of hereditary graph classes. A graph G is H-free if H is not an induced subgraph of G, and it is (H1,H2) -free if it is both H1 -free and H2 -free. A class of H-free graphs has bounded clique-width if and only if its atoms have this property. This is no longer true for (H1,H2) -free graphs, as evidenced by one known example. We prove the existence of another such pair (H1,H2) and classify the boundedness of clique-width on (H1,H2) -free atoms for all but 18 cases
Colouring (Pr+Ps)-free graphs
The k-Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for a fixed integer k such that no two adjacent vertices are coloured alike. If each vertex u must be assigned a colour from a prescribed list L(u) subseteq {1,...,k}, then we obtain the List k-Colouring problem. A graph G is H-free if G does not contain H as an induced subgraph. We continue an extensive study into the complexity of these two problems for H-free graphs. We prove that List 3-Colouring is polynomial-time solvable for (P_2+P_5)-free graphs and for (P_3+P_4)-free graphs. Combining our results with known results yields complete complexity classifications of 3-Colouring and List 3-Colouring on H-free graphs for all graphs H up to seven vertices. We also prove that 5-Colouring is NP-complete for (P_3+P_5)-free graphs
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