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The Complexity of the Shapley Value for Regular Path Queries

Authors: Majd Khalil and Benny Kimelfeld

Published in: LIPIcs, Volume 255, 26th International Conference on Database Theory (ICDT 2023)

A path query extracts vertex tuples from a labeled graph, based on the words that are formed by the paths connecting the vertices. We study the computational complexity of measuring the contribution of edges and vertices to an answer to a path query, focusing on the class of conjunctive regular path queries. To measure this contribution, we adopt the traditional Shapley value from cooperative game theory. This value has been recently proposed and studied in the context of relational database queries and has uses in a plethora of other domains. We first study the contribution of edges and show that the exact Shapley value is almost always hard to compute. Specifically, it is #P-hard to calculate the contribution of an edge whenever at least one (non-redundant) conjunct allows for a word of length three or more. In the case of regular path queries (i.e., no conjunction), the problem is tractable if the query has only words of length at most two; hence, this property fully characterizes the tractability of the problem. On the other hand, if we allow for an approximation error, then it is straightforward to obtain an efficient scheme (FPRAS) for an additive approximation. Yet, a multiplicative approximation is harder to obtain. We establish that in the case of conjunctive regular path queries, a multiplicative approximation of the Shapley value of an edge can be computed in polynomial time if and only if all query atoms are finite languages (assuming non-redundancy and conventional complexity limitations). We also study the analogous situation where we wish to determine the contribution of a vertex, rather than an edge, and establish complexity results of similar nature.

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Majd Khalil and Benny Kimelfeld. The Complexity of the Shapley Value for Regular Path Queries. In 26th International Conference on Database Theory (ICDT 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 255, pp. 11:1-11:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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  author =	{Khalil, Majd and Kimelfeld, Benny},
  title =	{{The Complexity of the Shapley Value for Regular Path Queries}},
  booktitle =	{26th International Conference on Database Theory (ICDT 2023)},
  pages =	{11:1--11:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-270-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{255},
  editor =	{Geerts, Floris and Vandevoort, Brecht},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-177535},
  doi =		{10.4230/LIPIcs.ICDT.2023.11},
  annote =	{Keywords: Path queries, regular path queries, graph databases, Shapley value}