104 research outputs found
On the maximum number of edges in planar graphs of bounded degree and matching number
We determine the maximum number of edges that a planar graph can have as a function of its maximum degree and matching number.publishedVersio
-Coloring Parameterized by Pathwidth is XNLP-complete
We show that the -Coloring problem is complete for the class XNLP when
parameterized by the pathwidth of the input graph. Besides determining the
precise parameterized complexity of this problem, this implies that b-Coloring
parameterized by pathwidth is -hard for all , and resolves the
parameterized complexity of -Coloring parameterized by treewidth
Structural Parameterizations of Clique Coloring
A clique coloring of a graph is an assignment of colors to its vertices such that no maximal clique is monochromatic. We initiate the study of structural parameterizations of the Clique Coloring problem which asks whether a given graph has a clique coloring with q colors. For fixed q ? 2, we give an ?^?(q^{tw})-time algorithm when the input graph is given together with one of its tree decompositions of width tw. We complement this result with a matching lower bound under the Strong Exponential Time Hypothesis. We furthermore show that (when the number of colors is unbounded) Clique Coloring is XP parameterized by clique-width
b-Coloring Parameterized by Clique-Width
We provide a polynomial-time algorithm for b-Coloring on graphs of constant clique-width. This unifies and extends nearly all previously known polynomial-time results on graph classes, and answers open questions posed by Campos and Silva [Algorithmica, 2018] and Bonomo et al. [Graphs Combin., 2009]. This constitutes the first result concerning structural parameterizations of this problem. We show that the problem is FPT when parameterized by the vertex cover number on general graphs, and on chordal graphs when parameterized by the number of colors. Additionally, we observe that our algorithm for graphs of bounded clique-width can be adapted to solve the Fall Coloring problem within the same runtime bound. The running times of the clique-width based algorithms for b-Coloring and Fall Coloring are tight under the Exponential Time Hypothesis
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)
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
XNLP-Completeness for Parameterized Problems on Graphs with a Linear Structure
In this paper, we showcase the class XNLP as a natural place for many hard problems parameterized by linear width measures. This strengthens existing W[1]-hardness proofs for these problems, since XNLP-hardness implies W[t]-hardness for all t. It also indicates, via a conjecture by Pilipczuk and Wrochna [ToCT 2018], that any XP algorithm for such problems is likely to require XP space.
In particular, we show XNLP-completeness for natural problems parameterized by pathwidth, linear clique-width, and linear mim-width. The problems we consider are Independent Set, Dominating Set, Odd Cycle Transversal, (q-)Coloring, Max Cut, Maximum Regular Induced Subgraph, Feedback Vertex Set, Capacitated (Red-Blue) Dominating Set, and Bipartite Bandwidth
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
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