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This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: In this paper, we study the complexity of several coloring problems on graphs, parameterized by the treewidth of the graph. 1. The List Coloring problem takes as input a graph G, together with an assignment to each vertex v of a set of colors C v. The problem is to determine whether it is possible to choose a color for vertex v from the set of permitted colors C v , for each vertex, so that the obtained coloring of G is proper. We show that this problem is W [1]-hard, parameterized by the treewidth of G. The closely related Precoloring Extension problem is also shown to be W [1]-hard, parameterized by treewidth. 2. An equitable coloring of a graph G is a proper coloring of the vertices where the numbers of vertices having any two distinct colors differs by at most one. We show that the problem is hard for W [1], parameterized by the treewidth plus the number of colors. We also show that a list-based variation, List Equitable Coloring is W [1]-hard for forests, parameter-ized by the number of colors on the lists. 3. 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 of one color from each vertex list L v yielding a proper coloring of G. We show that the problem of determining whether χ l (G) ≤ r, the List Chromatic Number problem, is solvable in linear time on graphs of constant treewidth.

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Clique-width is a graph parameter that measures in a certain sense the complexity of a graph. Hard graph problems (e.g., problems expressible in Monadic Second Order Logic with second-order quantification on vertex sets, that includes NP-hard problems such as 3-colorability) can be solved in polynomial time for graphs of bounded clique-width. We show that the clique-width of a given graph cannot be absolutely approximated in polynomial time unless P = NP. We also show that, given a graph G and an integer k, deciding whether the clique-width of G is at most k is NP-complete. This solves a problem that has been open since the introduction of clique-width in the early 1990s.

The computation of Kemeny rankings is central to many applications in the context of rank aggregation. Given a set of permutations (votes) over a set of candidates, one searches for a "consensus permutation" that is "closest" to the given set of permutations. Unfortunately, the problem is NP-hard. We provide a broad study of the parameterized complexity for computing optimal Kemeny rankings. Beside the three obvious parameters "number of votes", "number of candidates", and solution size (called Kemeny score), we consider further structural parameterizations. More specifically, we show that the Kemeny score (and a corresponding Kemeny ranking) of an election can be computed efficiently whenever the average pairwise distance between two input votes is not too large. In other words, Kemeny Score is fixedparameter tractable with respect to the parameter "average pairwise Kendall-Tau distance d a ". We describe a fixed-parameter algorithm with running time 16 ⌈da⌉ • poly. Moreover, we extend our studies to the parameters "maximum range" and "average range" of positions a candidate takes in the input votes. Whereas Kemeny Score remains fixed-parameter tractable with respect to the parameter "maximum range", it becomes NP-complete in case of an average range of two. This excludes fixed-parameter tractability with respect to the parameter "average range" unless P=NP. Finally, we extend some of our results to votes with ties and incomplete votes, where in both cases one no longer has permutations as input.

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