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Example of the reduction from MULTICOLOR CLIQUE to LIST COLORING.
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We study the complexity of several coloring problems on graphs, parameterized by the treewidth t of the graph:
(1) The list chromatic number
χ
l
(G) of a graph G is defined to be the smallest positive integer r, such that for every assignment to the vertices v of G, of a list L
v
of colors, where each list has length at least r, there is a choice o...
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Citations
... The complexity of WRP on bounded-treewidth graphs is of particular interest: while vertex-disjoint paths and cycles problems are often polynomial-time solvable on bounded treewidth graphs (e.g., vertex disjoint paths [55], vertex coloring, Hamiltonian cycles [7], Traveling Salesman [13], see also the works by Bodlaender [11] and Fellows et al. [28]) many edge-disjoint problem variants are -hard (e.g., edgedisjoint paths [50], edge coloring [48]). Moreover, the usual line graph construction approaches to transform vertex-disjoint to edge-disjoint problems are not applicable as such transformations do not preserve bounded treewidth. ...
We initiate the study of a fundamental combinatorial problem: Given a capacitated graph $G=(V,E)$, find a shortest walk ("route") from a source $s\in V$ to a destination $t\in V$ that includes all vertices specified by a set $\mathscr{W}\subseteq V$: the \emph{waypoints}. This waypoint routing problem finds immediate applications in the context of modern networked distributed systems. Our main contribution is an exact polynomial-time algorithm for graphs of bounded treewidth. We also show that if the number of waypoints is logarithmically bounded, exact polynomial-time algorithms exist even for general graphs. Our two algorithms provide an almost complete characterization of what can be solved exactly in polynomial-time: we show that more general problems (e.g., on grid graphs of maximum degree 3, with slightly more waypoints) are computationally intractable.
... The following result was proven implicitly in [9]. ...
... The proof in [9] relies on a reduction from Multicolored Clique [10] to Equitable coloring. The reduction transforms an instance of Multicolored clique of parameter k into an Equitable coloring instance of path-width and feedback vertex size at most O(k) (though only tree-width is explicitly stated in the paper). ...
Deletion problems are those where given a graph G and a graph property \(\pi \), the goal is to find a subset of edges such that after its removal the graph G will satisfy the property \(\pi \). Typically, we want to minimize the number of elements removed. In fair deletion problems we change the objective: we minimize the maximum number of deletions in a neighborhood of a single vertex. We study the parameterized complexity of fair deletion problems with respect to the structural parameters of the tree-width, the path-width, the size of a minimum feedback vertex set, the neighborhood diversity, and the size of minimum vertex cover of graph G. We prove the \(\mathsf {W[1]}\)-hardness of the fair \({{\mathsf {F}}}{{\mathsf {O}}}\) vertex-deletion problem with respect to the first three parameters combined. Moreover, we show that there is no algorithm for fair \({{\mathsf {F}}}{{\mathsf {O}}}\) vertex-deletion problem running in time \(n^{o(\root 3 \of {k})}\), where n is the size of the graph and k is the sum of the first three mentioned parameters, provided that the Exponential Time Hypothesis holds. On the other hand, we provide an FPT algorithm for the fair \(\mathsf {MSO}\) edge-deletion problem parameterized by the size of minimum vertex cover and an FPT algorithm for the fair \(\mathsf {MSO}\) vertex-deletion problem parameterized by the neighborhood diversity.
... We also recall that the partial k-trees are those graphs with treewidth at most k, for any k ≥ 1. Intuitively, the treewidth [33] is a parameter that measures how much a graph is similar to a tree, and it has applications in parameterized complexity (see, e.g., [16]). Since both types of colorability are expressible in the monadic second-order logic (MSO logic), they are decidable in linear time for graphs of bounded treewidth, as a consequence of Courcelle's theorem [7]. ...
Defective coloring is a variant of the traditional vertex-coloring in which adjacent vertices are allowed to have the same color, as long as the induced monochromatic components have a certain structure. Due to its important applications, as for example in the bipartisation of graphs, this type of coloring has been extensively studied, mainly with respect to the size, degree, diameter, and acyclicity of the monochromatic components. We focus on defective colorings with k colors in which the monochromatic components are acyclic and have small diameter, namely we consider (edge, k)-colorings, in which the monochromatic components have diameter 1, and (star, k)-colorings, in which they have diameter 2. We prove that the (edge, 3)-coloring problem remains NP-complete even for graphs with maximum vertex-degree 6, hence answering an open question posed by Cowen et al. [9], and for planar graphs with maximum vertex-degree 7, and we prove that the (star, 3)-coloring problem is NP-complete even for planar graphs with bounded maximum vertex-degree. On the other hand, we give linear-time algorithms for testing the existence of (edge, 2)-colorings and of (star, 2)-colorings of partial 2-trees. Finally, we prove that outerpaths, a notable subclass of outerplanar graphs, always admit (star, 2)-colorings.
... It is useful for proving lower bounds on NP-hard combinatorial problems . We follow a survey on this topic by Fellows et al. [9], which contains more details on this topic. ...
... Input: A graph G, an MSO formula φ with a free vertex-set variable, and a positive integer r ≥ 2. Question: Is there an equitable partition of vertices into r sets such that each class of the partition satisfies φ? Equitable MSO partition generalizes several problems already studied before. For example, if the formula φ(X) is " X is independent " , then we get an instance of Equitable coloring [9]. If we set φ(X) to " X is connected " , then we get an instance of Equitable connected partition [8]. ...
Edge deletion problems are those where given a graph G and a graph property $\pi$, the goal is to find a subset of edges such that after its removal the graph G will satisfy the property $\pi$. Typically, we want to minimize the number of edges removed. In fair deletion problem we change the objective: we minimize the maximum number of edges incident to a single vertex. We study the parameterized complexity of fair deletion problems with respect to the structural parameters of the tree-width, the path-width, the size of a minimum feedback vertex set, the neighborhood diversity, and the size of minimum vertex cover of graph G. We prove the W[1]-hardness of the fair MSO edge-deletion with respect to the first three parameters combined. Moreover, we show that there is no algorithm for fair MSO edge-deletion running in time $n^{o(\sqrt k)}$, where n is the size of the graph and k is the sum of the first three mentioned parameters, provided that the Exponential Time Hypothesis holds. On the other hand, we provide an FPT algorithm for the fair MSO edge-deletion parameterized by the size of minimum vertex cover and an FPT algorithm for the fair MSO vertex-deletion parameterized by the neighborhood diversity.
... We only show that the tree-decompositions of the graphs constructed there have height at most 3. The graph G ′ constructed in [13] consists of a disjoint union of trees of height 2. Hence, the treedecomposition of G ′ has height at most 3. The next step in their construction is to add an appropriate number of isolated vertices to G ′ . ...
It is known that a number of natural graph problems which are FPT parameterized by treewidth become W-hard when parameterized by clique-width. It is therefore desirable to find a different structural graph parameter which is as general as possible, covers dense graphs but does not incur such a heavy algorithmic penalty.
The main contribution of this paper is to consider a parameter called modular-width, defined using the well-known notion of modular decompositions. Using a combination of ILP and dynamic programming we manage to design FPT algorithms for Coloring and Partitioning into paths (and hence Hamiltonian path and Hamiltonian cycle), which are W-hard for both clique-width and its recently introduced restriction, shrub-depth. We thus argue that modular-width occupies a sweet spot as a graph parameter, generalizing several simpler notions on dense graphs but still evading the “price of generality” paid by clique-width.
... There are currently several general methods for classifying problems as FPT when the parameterization includes treewidth, such as tree-decomposition dynamic-programming[3,4,6]and Courcelle's Theorem[7]. Despite the general success of treewidth as a parameter that leads to FPT algorithms, there are still important problems that have no known FPT algorithm when treewidth is taken as the parameter (see e.g.[14]). It is therefore reasonable to explore the development of general methods for restricted subclasses of bounded treewidth graphs, such as graphs of bounded vertex cover number. ...
... Finally, Petkovšek showed that the class of k-letter graphs are well quasi ordered by induced subgraphs, for every fixed k ∈ N[39]. Our work is very much related to the study of parameterized problems using alternative, also known as the complexity ecology of parameters[13,14,36]. Some examples of results in this direction include FPT algorithms for graph layout problems parameterized by the vertex-cover[17], an algorithm for Graph Isomorphism parameterized by feedback-vertex-set[26], and a polynomial kernel for the Vertex Cover problem parameterized by feedback-vertex-set[22]. ...
We show that three subclasses of bounded treewidth graphs are well quasi ordered by refinements of the minor order. Specifically,
we prove that graphs with bounded vertex cover are well quasi ordered by the induced subgraph order, graphs with bounded feedback
vertex set are well quasi ordered by the topological-minor order, and graphs with bounded circumference are well quasi ordered
by the induced minor order. Our results give algorithms for recognizing any graph family in these classes which is closed
under the corresponding minor order refinement.
KeywordsWell quasi order–Treewidth–Tree decomposition–Bounded vertex cover graphs–Bounded feedback vertex set graphs–Bounded circumference graphs–Parameterized complexity–Exact algorithms–Meta-algorithms–Geometric graph recognition–String graphs
... Instance: A graph G and for each vertex v ∈ V (G) a list l(v) of allowed colors for v. Question: Is there a proper coloring for G where each vertex is colored with a color from its list? Theorem 1 ([10]). List Coloring is W[1]-hard when parameterized by the treewidth of the instance graph. ...
... The list-chromatic number or choice number of G is the smallest integer r such that G is r-list-colorable. Now, as shown by Fellows et al. [10], determining the list chromatic number of a given graph is fixed-parameter tractable when parameterized by the treewidth of the graph. ...
... One can fpt-reduce List Coloring to Precoloring Extension by encoding the lists by means of precolored vertices of degree one, without increasing the treewidth. Corollary 1 ([10]). Precoloring Extension is W[1]-hard when parameterized by the treewidth of the instance graph. ...
We consider combinatorial problems that can be solved in polynomial time for
graphs of bounded treewidth but where the order of the polynomial that bounds
the running time is expected to depend on the treewidth bound. First we review
some recent results for problems regarding list and equitable colorings,
general factors, and generalized satisfiability. Second we establish a new
hardness result for the problem of minimizing the maximum weighted outdegree
for orientations of edge-weighted graphs of bounded treewidth.
... [12]). It is known that there are several coloring problems such as Precoloring Extension and Equitable Coloring which are W[1]-hard when parameterized by treewidth, but fixed-parameter tractable parameterized by the vertex cover number [13,11]. These parameters also yield differences in the kernelization complexity of q- Coloring. ...
This paper studies the kernelization complexity of graph coloring problems
with respect to certain structural parameterizations of the input instances. We
are interested in how well polynomial-time data reduction can provably shrink
instances of coloring problems, in terms of the chosen parameter. It is well
known that deciding 3-colorability is already NP-complete, hence parameterizing
by the requested number of colors is not fruitful. Instead, we pick up on a
research thread initiated by Cai (DAM, 2003) who studied coloring problems
parameterized by the modification distance of the input graph to a graph class
on which coloring is polynomial-time solvable; for example parameterizing by
the number k of vertex-deletions needed to make the graph chordal. We obtain
various upper and lower bounds for kernels of such parameterizations of
q-Coloring, complementing Cai's study of the time complexity with respect to
these parameters.
Our results show that the existence of polynomial kernels for q-Coloring
parameterized by the vertex-deletion distance to a graph class F is strongly
related to the existence of a function f(q) which bounds the number of vertices
which are needed to preserve the NO-answer to an instance of q-List-Coloring on
F.
... An instance of p-Pw-Sat problem is a triple (F , part : Φ → [k], tg : [k] → N), where F is a propositional CNF formula, part partitions the set of propositional variables into k parts and we need to check if there is a satisfying assignment that sets exactly tg(p) variables to ⊤ in each part p. Parameters are k and pathwidth of the primal graph of F (one vertex for each propositional variable, an edge between two variables iff they occur together in a clause). The following lemma can be proved by a Fpt reduction from the Number List Coloring Problem [11]. ...
... [11], it is proved that even for graphs of pathwidth 2, Nlcp is W[1]-hard when parameterized by total number of colors in ∪ v∈V S v . ...
Many tractable algorithms for solving the Constraint Satisfaction Problem (Csp) have been developed using the notion of the treewidth of some graph derived from the input Csp instance. In particular, the incidence graph of the Csp instance is one such graph. We introduce the notion of an incidence graph for modal logic formulae in a certain normal form.
We investigate the parameterized complexity of modal satisfiability with the modal depth of the formula and the treewidth
of the incidence graph as parameters. For various combinations of Euclidean, reflexive, symmetric and transitive models, we
show either that modal satisfiability is Fpt, or that it is W[1]-hard. In particular, modal satisfiability in general models is Fpt, while it is W[1]-hard in transitive models. As might be expected, modal satisfiability in transitive and Euclidean models
is Fpt.
... In other words, the task is to extend a partial k-coloring to the entire graph. It has been shown that on general graphs PrExt is not fixed-parameter tractable with respect to parameter treewidth [10] but, other than List Coloring, it becomes fixed-parameter tractable when parameterized by vertex cover size [13]. Moreover, it is NP-complete for fixed k ≥ 3 on planar bipartite graphs [15] and W [1]-hard with respect to the number of precolored vertices for chordal graphs [16]. ...
... Clearly, G i is k-colorable iff G i can be incrementally colored by recoloring at most c vertices in a k-coloring for G i−1 . Thus, we can decide the question about the k-colorability of G inductively by deciding at most n recoloring problems, implying the NP-hardness of IC k-List Coloring for k ≥ 3. Using the above proof strategy, the result that k-List Coloring is W [1] -hard with respect to the parameter treewidth [10] can be transferred to IC k-List Coloring. Moreover, it also follows that IC k-List Coloring is NP-complete for fixed k ≥ 3 for all hereditary graph classes 1 where the ordinary List Coloring problem is NP-hard for fixed k ≥ 3, e. g., planar bipartite and chordal graphs. ...
... However, it is W [1]-hard, again excluding hope for fixed-parameter tractability. In order to show the W [1]-hardness, we present a parameterized reduction from the W [1]-complete k-Multicolored Independent Set problem [10, 11]. The problem is to decide for a given k-coloring f for a graph G = (V, E) whether there exists a multicolored k-independent set, that is, a vertex subset S ⊆ V with |S| = k such that ∀u, v ∈ S : {u, v} E ∧ f (u) f (v). ...
Incrementally k-list coloring a graph means that a graph is given by adding stepwise one vertex after another, and for each intermediate
step we ask for a vertex coloring such that each vertex has one of the colors specified by its associated list containing
some of in totalk colors. We introduce the “conservative version” of this problem by adding a further parameterc ∈ ℕ specifying the maximum number of vertices to be recolored between two subsequent graphs (differing by one vertex). This
“conservation parameter”c models the natural quest for a modest evolution of the coloring in the course of the incremental process instead of performing
radical changes. We show that the problem is NP-hard for k ≥ 3 and W[1]-hard when parameterized byc. In contrast, the problem becomes fixed-parameter tractable with respect to the combined parameter(k,c). We prove that the problem has an exponential-size kernel with respect to(k,c) and there is no polynomial-size kernel unless NP ⊆ coNP/poly. Finally, we provide empirical findings for the practical relevance of our approach in terms of an effective graph coloring
heuristic.
