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<Paper uid="W01-0805">
  <Title>A Meta-Algorithm for the Generation of Referring Expressions</Title>
  <Section position="3" start_page="0" end_page="0" type="intro">
    <SectionTitle>
2 Graphs
</SectionTitle>
    <Paragraph position="0"> In this scene, as in any other scene, we see a finite set of entities a12 with properties a13 and relations a14 . In this particular scene, the set a12a16a15a18a17a20a19a22a21a24a23a25a19a27a26a6a23a25a19a29a28a4a23a25a19a29a30a32a31 is the set of entities, a13a33a15 a17 dog, chihuahua, doghouse, small, large, white, brown a31 is the set of properties and a14a34a15a35a17 next to, left of, right of, contain, in a31 is the set of relations.</Paragraph>
    <Paragraph position="1"> A scene can be represented in various ways. One common representation is to build a database, listing the properties of each element of a12 :</Paragraph>
    <Paragraph position="3"> Here we take a different approach and represent a scene as a labeled directed graph. Let a40a41a15a42a13a44a43a45a14 be the set of labels (with a13 and a14 disjoint, i.e., a13a47a46a48a14 a15 a49 ). Then, a labeled directed graph  a15a52a51a54a53a55a23a57a56a59a58 , where a53a61a60a62a12 is the set of vertices (or nodes) and a56a63a60a64a53a66a65a67a40a68a65a69a53 is the set of labeled directed arcs (or edges). The scene given in Figure 1 can be represented by the graph in Figure 2.</Paragraph>
    <Paragraph position="4"> Keep in mind that the a19 labels are only added to ease reference to nodes. Notice also that properties (such as being a dog) are always modelled as loops, i.e., edges which start and end in the same node, while relations may (but need not) have different start and end nodes.</Paragraph>
    <Paragraph position="5"> Now the content determination problem for referring expressions can be formulated as a graph construction task. In order to decide which information to include in a referring expression for an object a19a63a70a71a53 , we construct a connected directed labeled graph over the set of labels a40 and an arbitrary set of nodes, but including a19 . This graph can be understood as the &amp;quot;meaning representation&amp;quot; from which a referring expression can be generated by a linguistic realizer. Informally, we say that a graph refers to a given entity iff the graph can be &amp;quot;placed over&amp;quot; the scene graph in such a way that the node being referred to is &amp;quot;placed over&amp;quot; the given entity and each edge can be &amp;quot;placed over&amp;quot; an edge labeled with the same label. Furthermore, a graph is distinguishing iff it refers to exactly one node in the scene graph.</Paragraph>
    <Paragraph position="6"> Consider the three graphs in Figure 3. Here and elsewhere circled nodes stand for the intended referent. Graph (i) refers to all nodes of the graph in Figure 2 (every object in the scene is next to some other object), graph (ii) can refer to both a19a72a21 and a19a27a26 , and graph (iii) is distinguishing in that it can only refer to a19a72a21 . Notice that the three graphs might be realized as something next to something else, a chihuahua and the dog in the doghouse respectively. In this paper, we will concentrate on the generation of distinguishing graphs.</Paragraph>
    <Paragraph position="7"> Formally, the notion that a graph a73 a15 a51a54a53a75a74a76a23a57a56a77a74a78a58 can be &amp;quot;placed over&amp;quot; another graph  a15a79a51a54a53a81a80a82a23a57a56a77a80a55a58 corresponds to the notion of a sub-graph isomorphism. a73 can be &amp;quot;placed over&amp;quot; a50 iff there exists a subgraph a50a77a83 a15a79a51a54a53 a80a85a84 a23a57a56 a80a85a84 a58 of a50 such that a73 is isomorphic to a50 a83 . a73 is isomorphic to a50 a83 iff there exists a bijection a86a88a87a89a53 a74a91a90 a53 a80a85a84 such that for all nodes a92a93a23a57a94a63a70a95a53 a74 and all a96a97a70a98a40</Paragraph>
    <Paragraph position="9"> In words: the bijective function a86 maps all the nodes in a73 to corresponding nodes in a50 a83 , in such  with circles around the intended referent. and a94 in a73 is matched by an edge with the same label between the a50 a83 counterparts of a92 and a94 , i.e., a86a105a104a106a92 and a86a105a104a106a94 respectively. When a73 is isomorphic to some subgraph of a50 by an isomorphism a86 , we write a73a127a126a129a128 a50 .</Paragraph>
    <Paragraph position="10"> Given a graph a73 and a node a92 in a73 , and a graph a50 and a node a94 in a50 , we define that the pair a99 a92a93a23a57a73a130a101 refers to the pair a99 a94a76a23 a50 a101 iff a73 is connected and</Paragraph>
    <Paragraph position="12"> uniquely refers to a99 a94a45a23 a50 a101 (i.e., a99 a92a93a23a57a73a130a101 is distinguishing) iff a99 a92a93a23a57a73a130a101 refers to a99 a94a76a23 a50 a101 and there is no node a94 a83 in a50 different from a94 such that a99 a92a93a23a57a73a130a101 refers to a99 a94 a83 a23 a50 a101 . The problem considered in this paper can now be formalized as follows: given a graph a50 and a node a94 in a50 , find a pair a99 a92a81a23a57a73a132a101 such that a99 a92a93a23a57a73a130a101 uniquely refers to a99 a94a76a23 a50 a101 .</Paragraph>
    <Paragraph position="13"> Consider, for instance, the task of finding a pair  a92a81a23a57a73a132a101 which uniquely refers to the node labeled a19 a21 in Figure 2. It is easily seen that there are a number of such pairs, three of which are depicted in Figure 4. We would like to have a mechanism which allows us to give certain solutions preference over other solutions. For this purpose we shall use cost-functions. In general, a cost function a133 is a function which assigns to each sub-graph of a scene graph a positive number. As we shall see, by defining cost functions in different ways, we can mimic various algorithms for the generation of referring expressions known from the literature.</Paragraph>
    <Paragraph position="14"> A note on problem complexity The basic decision problem for subgraph isomorphism (i.e., testing whether a graph a73 is isomorphic to a sub-graph of a50 ) is known to be NP complete (see e.g., Garey &amp; Johnson 1979). Here we are interested in connected a73 , but unfortunately that  restriction does not reduce the theoretical complexity. However, as soon as we define an upper bound a146 on the number of edges in a distinguishing graph, the problem loses its intractability and becomes solvable in polynomial a147 a99a149a148a151a150 a101 time. Such a restriction is rather harmless for our current purposes, as it would only prohibit the generation of distinguishing descriptions with more than a146 properties, for an arbitrary large a146 . In general, there are various classes of graphs for which the subgraph isomorphism problem can be solved much more efficiently, without postulating upper bounds. For instance, if a50 and a73 are planar graphs the problem can be solved in time linear in the number of nodes of a50 (Eppstein 1999). Basically, a planar graph is one which can be drawn on a plane in such a way that there are no crossing edges (thus, for instance, the graph in Figure 2 is planar). It is worth investigating to what extent planar graphs suffice for the generation of referring expressions.</Paragraph>
  </Section>
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