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<Paper uid="P00-1047">
  <Title>A Polynomial-Time Fragment of Dominance Constraints</Title>
  <Section position="9" start_page="0" end_page="0" type="concl">
    <SectionTitle>
7 Conclusion
</SectionTitle>
    <Paragraph position="0"> We have investigated normal dominance constraints, a natural subclass of general dominance constraints. We have given an O(n2 log n) satis ability algorithm for them and integrated it into an algorithm that enumerates all irredundant solved forms in time O(Nn4 log n), where N is the number of irredundant solved forms.</Paragraph>
    <Paragraph position="1"> 5A constructive solution is one where every node in the model is the image of a variable for which a labeling literal is in the constraint. Informally, this means that the solution only contains \material&amp;quot; \mentioned&amp;quot; in the constraint.</Paragraph>
    <Paragraph position="2"> This eliminates any doubts about the computational practicability of dominance constraints which were raised by the NP-completeness result for the general language (Koller et al., 1998) and expressed e.g. in (Willis and Manandhar, 1999). First experiments con rm the e ciency of the new algorithm { it is superior to the NP algorithms especially on larger constraints.</Paragraph>
    <Paragraph position="3"> On the other hand, we have argued that the problem of nding constructive solutions even of a normal dominance constraint is NPcomplete. This result carries over to other underspeci cation formalisms, such as Hole Semantics and MRS. In practice, however, it seems that the enumeration algorithm presented here can be adapted to those problems. Acknowledgments. We would like to thank Ernst Althaus, Denys Duchier, Gert Smolka, Sven Thiel, all members of the SFB 378 project CHORUS at the University of the Saarland, and our reviewers. This work was supported by the DFG in the SFB 378.</Paragraph>
  </Section>
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