Abstract
We consider the #hbox {P}-complete problem of counting the number of linear extensions of a poset (textsc {#LE}); a fundamental problem in order theory with applications in a variety of distinct areas. In particular, we study the complexity of textsc {#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 textsc {#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 {textsc {#LE}} becomes fixed-parameter tractable parameterized by the treewidth of the incomparability graph.
Highlights
Counting the number of linear extensions of a poset is a fundamental problem of order theory that has applications in a variety of distinct areas such as sorting [30], sequence analysis [25], convex rank tests [27], sampling schemes of Bayesian networks [28], and preference reasoning [24]
In this paper we study the complexity of counting linear extensions when the parameter is the treewidth—a fundamental graph parameter which has already found a plethora applications in many areas of computer science [17,18,29]
We settle the fixed-parametertractability of the problem when parameterizing by the treewidth of two of the most prominent graphical representations of posets, the cover graph and the incomparability graph
Summary
Counting the number of linear extensions of a poset is a fundamental problem of order theory that has applications in a variety of distinct areas such as sorting [30], sequence analysis [25], convex rank tests [27], sampling schemes of Bayesian networks [28], and preference reasoning [24]. The exact dynamic programming algorithm [10] can be shown to run in time O(nw · w) for a poset with n elements and width w (the size of the largest anti-chain) None of these efforts has so far led to an fpt algorithm. We believe that this uncertainty about the exact complexity status of counting linear extensions with respect to these various parameterizations is at least partly due to the fact that we deal with a counting problem whose decision version is trivial, i.e., every poset has at least one linear extension. The same predicament makes studying the complexity of counting linear extensions significantly more interesting, as noted by Flum and Grohe [16]: The theory gets interesting with those counting problems that are harder than their corresponding decision versions
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