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<Paper uid="P04-1051">
  <Title>Computing Locally Coherent Discourses</Title>
  <Section position="2" start_page="0" end_page="0" type="intro">
    <SectionTitle>
1 Introduction
</SectionTitle>
    <Paragraph position="0"> One central problem in discourse generation and summarisation is to structure the discourse in a way that maximises coherence. Coherence is the property of a good human-authored text that makes it easier to read and understand than a randomlyordered collection of sentences.</Paragraph>
    <Paragraph position="1"> Several papers in the recent literature (Mellish et al., 1998; Barzilay et al., 2002; Karamanis and Manurung, 2002; Lapata, 2003; Karamanis et al., 2004) have focused on defining local coherence, which evaluates the quality of sentence-to-sentence transitions. This is in contrast to theories of global coherence, which can consider relations between larger chunks of the discourse and e.g. structures them into a tree (Mann and Thompson, 1988; Marcu, 1997; Webber et al., 1999). Measures of local coherence specify which ordering of the sentences makes for the most coherent discourse, and can be based e.g.</Paragraph>
    <Paragraph position="2"> on Centering Theory (Walker et al., 1998; Brennan et al., 1987; Kibble and Power, 2000; Karamanis and Manurung, 2002) or on statistical models (Lapata, 2003).</Paragraph>
    <Paragraph position="3"> But while formal models of local coherence have made substantial progress over the past few years, the question of how to efficiently compute an ordering of the sentences in a discourse that maximises local coherence is still largely unsolved. The fundamental problem is that any of the factorial number of permutations of the sentences could be the optimal discourse, which makes for a formidable search space for nontrivial discourses. Mellish et al. (1998) and Karamanis and Manurung (2002) present algorithms based on genetic programming, and Lapata (2003) uses a graph-based heuristic algorithm, but none of them can give any guarantees about the quality of the computed ordering.</Paragraph>
    <Paragraph position="4"> This paper presents the first algorithm that computes optimal locally coherent discourses, and establishes the complexity of the discourse ordering problem. We first prove that the discourse ordering problem for local coherence measures is equivalent to the Travelling Salesman Problem (TSP). This means that discourse ordering is NP-complete, i.e.</Paragraph>
    <Paragraph position="5"> there are probably no polynomial algorithms for it.</Paragraph>
    <Paragraph position="6"> Worse, our result implies that the problem is not even approximable; any polynomial algorithm will compute arbitrarily bad solutions on unfortunate inputs. Note that all approximation algorithms for the TSP assume that the underlying cost function is a metric, which is not the case for the coherence measures we consider.</Paragraph>
    <Paragraph position="7"> Despite this negative result, we show that by applying modern algorithms for TSP, the discourse ordering problem can be solved efficiently enough for practical applications. We define a branch-and-cut algorithm based on linear programming, and evaluate it on discourse ordering problems based on the GNOME corpus (Karamanis, 2003) and the BLLIP corpus (Lapata, 2003). If the local coherence measure depends only on the adjacent pairs of sentences in the discourse, we can order discourses of up to 50 sentences in under a second. If it is allowed to depend on the left-hand context of the sentence pair, computation is often still efficient, but can become expensive.</Paragraph>
    <Paragraph position="8"> The structure of the paper is as follows. We will first formally define the discourse ordering problem and relate our definition to the literature on local coherence measures in Section 2. Then we will prove the equivalence of discourse ordering and TSP (Section 3), and present algorithms for solving it in Section 4. Section 5 evaluates our algorithms on examples from the literature. We compare our approach to various others in Section 6, and then conclude in Section 7.</Paragraph>
  </Section>
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