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<Paper uid="P04-1051">
  <Title>Computing Locally Coherent Discourses</Title>
  <Section position="8" start_page="0" end_page="0" type="concl">
    <SectionTitle>
7 Conclusion
</SectionTitle>
    <Paragraph position="0"> We have shown that the problem of ordering clauses into a discourse that maximises local coherence is equivalent to the travelling salesman problem: Even the two-place discourse ordering problem can encode ATSP. This means that the problem is NP-complete and doesn't even admit polynomial approximation algorithms (unless P=NP).</Paragraph>
    <Paragraph position="1"> On the other hand, we have shown how to encode the discourse ordering problems of arbitrary arity d into GATSP. We have demonstrated that modern branch-and-cut algorithms for GATSP can easily solve practical discourse ordering problems if d = 2, and are still usable for many instances with d = 3. As far as we are aware, this is the first algorithm for discourse ordering that can make any guarantees about the solution it computes.</Paragraph>
    <Paragraph position="2"> Our efficient implementation can benefit generation and summarisation research in at least two respects. First, we show that computing locally coherent orderings of clauses is feasible in practice, as such coherence measures will probably be applied on sentences within the same paragraph, i.e.</Paragraph>
    <Paragraph position="3"> on problem instances of limited size. Second, our system should be a useful experimentation tool in developing new measures of local coherence.</Paragraph>
    <Paragraph position="4"> We have focused on local coherence in this paper, but it seems clear that notions of global coherence, which go beyond the level of sentence-to-sentence transitions, capture important aspects of coherence that a purely local model cannot. However, our algorithm can still be useful as a subroutine in a more complex system that deals with global coherence (Marcu, 1997; Mellish et al., 1998). Whether our methods can be directly applied to the tree structures that come up in theories of global coherence is an interesting question for future research.</Paragraph>
    <Paragraph position="5"> Acknowledgments. We would like to thank Mirella Lapata for providing the experimental data and Andrea Lodi for providing an efficiency base-line by running his ATSP solver on our inputs. We are grateful to Malte Gabsdil, Ruli Manurung, Chris Mellish, Kristina Striegnitz, and our reviewers for helpful comments and discussions.</Paragraph>
  </Section>
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