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Exact exponential algorithms for two poset problems

Partially ordered sets (posets) are fundamental combinatorial objects with important applications in computer science. Perhaps the most natural algorithmic task, given a size-$n$ poset, is to compute its number of linear extensions. In 1991 Brightwell and Winkler showed this problem to be $\#P$-hard. In spite of extensive research, the fastest known algorithm is still the straightforward $O(n 2^n)$-time dynamic programming (an adaptation of the Bellman-Held-Karp algorithm for the TSP). Very recently, Dittmer and Pak showed that the problem remains $\#P$-hard for two-dimensional posets, and no algorithm was known to break the $2^n$-barrier even in this special case. The question of whether the two-dimensional problem is easier than the general case was raised decades ago by Möhring, Felsner and Wernisch, and others. In this paper we show that the number of linear extensions of a two-dimensional poset can be computed in time $O(1.8172^n)$. The related jump number problem asks for a linear extension of a poset, minimizing the number of neighboring incomparable pairs. The problem has applications in scheduling, and has been widely studied. In 1981 Pulleyblank showed it to be NP-complete. We show that the jump number problem can be solved (in arbitrary posets) in time $O(1.824^n)$. This improves (slightly) the previous best bound of Kratsch and Kratsch.