In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.

In many combinatorial problems one may need to model the diversity or similarity of sets of assignments. For example, one may wish to maximise or minimise the number of distinct values in a solution. To formulate problems of this type we can use soft variants of the well known AllDifferent and AllEqual constraints. We present a taxonomy of six soft global constraints, generated by combining the two latter ones and the two standard cost functions, which are either maximised or minimised. We characterise the complexity of achieving arc and bounds consistency on these constraints, resolving those cases for which NP-hardness was neither proven nor disproven. In particular, we explore in depth the constraint ensuring that at least k pairs of variables have a common value. We show that achieving arc consistency is NP-hard, however bounds consistency can be achieved in polynomial time through dynamic programming. Moreover, we show that the maximum number of pairs of equal variables can be approximated by a factor of 12 with a linear time greedy algorithm. Finally, we provide a fixed parameter tractable algorithm with respect to the number of values appearing in more than two distinct domains. Interestingly, this taxonomy shows that enforcing equality is harder than enforcing difference.
Soft Constraints of Difference and Equality

We present a method for solving implicit (factored) Markov decision processes (MDPs) with very large state spaces. We intro­ duce a property of state space partitions which we call f-homogeneity. Intuitively, an f-homogeneous partition groups together states that behave approximately the same under all or some subset of policies. Borrow­ ing from recent work on model minimization in computer-aided software verification, we present an algorithm that takes a factored representation of an MDP and an 0 � f � I and computes a factored f-homogeneous par­ tition of the state space. This partition defines a family of related MOPs-those MOP's with state space equal to the blocks of the partition, and transition probabilities "appro:X:imately" like those of any (original MDP) state in the source block. To formally study such families of MDPs, we introduce the new notion of a "bounded parameter MDP" (BMDP), which is a fam­ ily of (traditional) MOPs defined by speci­ fying upper and lower bounds on the transi­ tion probabilities and rewards. We describe algorithms that operate on BMDPs to find policies that are approximately optimal with respect to the original MDP. In combination, our method for reducing a large implicit MDP to a possibly much smaller BMDP using an f-homogeneous par­ tition, and our methods for selecting actions in BMDP's constitute a new approach for an­ alyzing large implicit MOP's. Among its ad­ vantages, this new approach provides insight into existing algorithms to solving implicit MDPs, provides useful connections to work in automata theory and model minimization, and suggests methods, which involve vary­ ing f, to trade time and space (specifically in terms of the size of the corresponding state space) for solution quality.
Model Reduction Techniques for Computing Approximately Optimal Solutions for Markov Decision Processes

Neural word representations have proven useful in Natural Language Processing (NLP) tasks due to their ability to efficiently model complex semantic and syntactic word relationships. However, most techniques model only one representation per word, despite the fact that a single word can have multiple meanings or ”senses”. Some techniques model words by using multiple vectors that are clustered based on context. However, recent neural approaches rarely focus on the application to a consuming NLP algorithm. Furthermore, the training process of recent word-sense models is expensive relative to single-sense embedding processes. This paper presents a novel approach which addresses these concerns by modeling multiple embeddings for each word based on supervised disambiguation, which provides a fast and accurate way for a consuming NLP model to select a sense-disambiguated embedding. We demonstrate that these embeddings can disambiguate both contrastive senses such as nominal and verbal senses as well as nuanced senses such as sarcasm. We further evaluate Part-of-Speech disambiguated embeddings on neural dependency parsing, yielding a greater than 8% average error reduction in unlabeled attachment scores across 6 languages.

A FAST AND ACCURATE METHOD FOR WORD SENSE DISAMBIGUATION IN NEURAL WORD EMBEDDINGS

