Algorithms for the Generalized Poset Sorting Problem
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We consider a generalized poset sorting problem (GPS), in which we are given a query graph G = (V, E) and an unknown poset 𝒫(V, ≺) that is defined on the same vertex set V, and the goal is to make as few queries as possible to edges in G in order to fully recover 𝒫, where each query (u, v) returns the relation between u, v, i.e., u ≺ v, v ≺ u or u ≁v. This generalizes both the poset sorting problem [Faigle et al., SICOMP 88] and the generalized sorting problem [Huang et al., FOCS 11]. We give algorithms with Õ(n·poly(k)) query complexity when G is a complete bipartite graph or G is stochastic under the model, where k is the width of the poset, and these generalize [Daskalakis et al., SICOMP 11] which only studies complete graph G. Both results are based on a unified framework that reduces the poset sorting to partitioning the vertices with respect to a given pivot element, which may be of independent interest. Our study of GPS also leads to a new Õ(n^1 - 1 / (2W)) competitive ratio for the so-called weighted generalized sorting problem where W is the number of distinct weights in the query graph. This problem was considered as an open question in [Charikar et al., JCSS 02], and our result makes important progress as it yields the first nontrivial Õ(n) ratio for general weighted query graphs (and better ratio if W is bounded). We obtain this via an Õ(nk + n^1.5) query complexity algorithm for the case where every edge in G is guaranteed to be comparable in the poset, which generalizes the state-of-the-art Õ(n^1.5) bound for generalized sorting [Huang et al., FOCS 11].
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