Mining Posets from Linear Orders

There has been much research on the combinatorial problem of generating the linear extensions of a given poset. This paper focuses on the reverse of that problem, where the input is a set of linear orders, and the goal is to construct a poset or set of posets that generates the input. Such a problem finds applications in computational neuroscience, systems biology, paleontology, and physical plant engineering. In this paper, several algorithms are presented for efficiently finding a single poset that generates the input set of linear orders. The variation of the problem where a minimum set of posets that cover the input is also explored. It is found that the problem is polynomially solvable for one class of simple posets (kite(2) posets) but NP-complete for a related class (hammock(2,2,2) posets)

Approximation and Computational Complexity of Some Hammock Variations of the Poset Cover Problem

The Hammock(⏟, , … , / )-Poset Cover Problem is a variation of the Poset Cover Problem with the same input – set {, , … , } of linear orders over the set {, , … ,}, but the solution is restricted to a set of simple hammock(⏟, , … , / ) posets. The problem is NP-Hard when ≥ but is in when = . The computational complexity of the problem when = is not yet known. In this paper, we determine the approximation complexity of the cases that have been shown to be NP-Hard. We show that the Hammock(⏟, , … , / )-Poset Cover Problem is in APX and, in particular, ( + / )-approximable, for ≥ . On the other hand, we also explore the computational complexity for the case where = [Hammock(2,2)-Poset Cover Problem]. We show that it is in when the transposition graph of the input set of linear orders is rectangula

Deep Unsupervised Similarity Learning using Partially Ordered Sets

Unsupervised learning of visual similarities is of paramount importance to computer vision, particularly due to lacking training data for fine-grained similarities. Deep learning of similarities is often based on relationships between pairs or triplets of samples. Many of these relations are unreliable and mutually contradicting, implying inconsistencies when trained without supervision information that relates different tuples or triplets to each other. To overcome this problem, we use local estimates of reliable (dis-)similarities to initially group samples into compact surrogate classes and use local partial orders of samples to classes to link classes to each other. Similarity learning is then formulated as a partial ordering task with soft correspondences of all samples to classes. Adopting a strategy of self-supervision, a CNN is trained to optimally represent samples in a mutually consistent manner while updating the classes. The similarity learning and grouping procedure are integrated in a single model and optimized jointly. The proposed unsupervised approach shows competitive performance on detailed pose estimation and object classification.Comment: Accepted for publication at IEEE Computer Vision and Pattern Recognition 201

On the Complexity of Mining Itemsets from the Crowd Using Taxonomies

We study the problem of frequent itemset mining in domains where data is not recorded in a conventional database but only exists in human knowledge. We provide examples of such scenarios, and present a crowdsourcing model for them. The model uses the crowd as an oracle to find out whether an itemset is frequent or not, and relies on a known taxonomy of the item domain to guide the search for frequent itemsets. In the spirit of data mining with oracles, we analyze the complexity of this problem in terms of (i) crowd complexity, that measures the number of crowd questions required to identify the frequent itemsets; and (ii) computational complexity, that measures the computational effort required to choose the questions. We provide lower and upper complexity bounds in terms of the size and structure of the input taxonomy, as well as the size of a concise description of the output itemsets. We also provide constructive algorithms that achieve the upper bounds, and consider more efficient variants for practical situations.Comment: 18 pages, 2 figures. To be published to ICDT'13. Added missing acknowledgemen

Mining Mixed-Initiative Dialogs

Human-computer dialogs are an important vehicle through which to produce a rich and compelling form of human-computer interaction. We view the specification of a human-computer dialog as a set of sequences of progressive interactions between a user and a computer system, and mine partially ordered sets, which correspond to mixing dialog initiative, embedded in these sets of sequences—a process we refer to as dialog mining—because partially ordered sets can be advantageously exploited to reduce the control complexity of a dialog implementation. Our mining losslessly compresses the specification of a dialog. We describe our mining algorithm and report the results of a simulation-oriented evaluation. Our algorithm is sound, and our results indicate that it can compress nearly all dialog specifications, and some to a high degree. This work is part of broader research on the specification and implementation of mixed-initiative dialogs

Diversity as Width

It is argued that if the population of options is a finite poset, diversity comparisons may be conveniently based on widths i.e. on the respective maximum numbers of pairwise incomparable options included in the relevant subposets. The width-ranking and the undominated width-ranking are introduced and characterized

A faster tree-decomposition based algorithm for counting linear extensions

We consider the problem of counting the linear extensions of an n-element poset whose cover graph has treewidth at most t. We show that the problem can be solved in time Õ(nt+3), where Õ suppresses logarithmic factors. Our algorithm is based on fast multiplication of multivariate polynomials, and so differs radically from a previous Õ(nt+4)-time inclusion–exclusion algorithm. We also investigate the algorithm from a practical point of view. We observe that the running time is not well characterized by the parameters n and t alone, fixing of which leaves large variance in running times due to uncontrolled features of the selected optimal-width tree decomposition. For selecting an efficient tree decomposition we adopt the method of empirical hardness models, and show that it typically enables picking a tree decomposition that is significantly more efficient than a random optimal-width tree decomposition. © Kustaa Kangas, Mikko Koivisto, and Sami Salonen; licensed under Creative Commons License CC-BY.Peer reviewe