Counting Linear Extensions: Parameterizations by Treewidth
نویسندگان
چکیده
We consider the #P-complete problem of counting the number of linear extensions of a poset (#LE); a fundamental problem in order theory with applications in a variety of distinct areas. In particular, we study the complexity of #LE parameterized by the well-known decompositional parameter treewidth for two natural graphical representations of the input poset, i.e., the cover and the incomparability graph. Our main result shows that #LE is fixed-parameter intractable parameterized by the treewidth of the cover graph. This resolves an open problem recently posed in the Dagstuhl seminar on Exact Algorithms. On the positive side we show that #LE becomes fixed-parameter tractable parameterized by the treewidth of the incomparability graph. 1998 ACM Subject Classification F.1.3 Complexity Measures and Classes, G.2.1 Combinatorics
منابع مشابه
Counting Linear Extensions of Sparse Posets
Counting the linear extensions of a partially ordered set (poset) is a fundamental problem with several applications. We present two exact algorithms that target sparse posets in particular. The first algorithm breaks the counting task into subproblems recursively. The second algorithm uses variable elimination via inclusion–exclusion and runs in polynomial time for posets with a cover graph of...
متن کاملProbabilistic Inference and Monadic Second Order Logic
This paper combines two classic results from two different fields: the result by Lauritzen and Spiegelhalter [21] that the probabilistic inference problem on probabilistic networks can be solved in linear time on networks with a moralization of bounded treewidth, and the result by Courcelle [10] that problems that can be formulated in counting monadic second order logic can be solved in linear ...
متن کاملGraph Layout Problems Parameterized by Vertex Cover
In the framework of parameterized complexity, one of the most commonly used structural parameters is the treewidth of the input graph. The reason for this is that most natural graph problems turn out to be fixed parameter tractable when parameterized by treewidth. However, Graph Layout problems are a notable exception. In particular, no fixed parameter tractable algorithms are known for the Cut...
متن کاملTree-width and Logspace: Determinants and Counting Euler Tours
Motivated by the recent result of [EJT10] showing that MSO properties are Logspace computable on graphs of bounded tree-width, we consider the complexity of computing the determinant of the adjacency matrix of a bounded tree-width graph and prove that it is L-complete. It is important to notice that the determinant is neither an MSO-property nor counts the number of solutions of an MSO-predicat...
متن کاملSimplified Algorithmic Metatheorems Beyond MSO: Treewidth and Neighborhood Diversity
This paper settles the computational complexity of model checking of several extensions of the monadic second order (MSO) logic on two classes of graphs: graphs of bounded treewidth and graphs of bounded neighborhood diversity. A classical theorem of Courcelle states that any graph property definable in MSO is decidable in linear time on graphs of bounded treewidth. Algorithmic metatheorems lik...
متن کامل