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arXiv 2212.07720cs.DB
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正则路图Shapley值的复杂性

The Complexity of the Shapley Value for Regular Path Queries

Majd Khalil, Benny Kimelfeld

摘要:

路径查询基于由连接顶点的路径形成的字从标记图中提取顶点元组。我们研究的计算复杂性测量的贡献的边和顶点的回答的路径查询,集中在类合取规则的路径查询。为了衡量这种贡献,我们采用了合作博弈理论中传统的Shapley值。这个值最近在关系数据库查询的上下文中被提出和研究,并在许多其他领域中使用。我们首先研究了边缘的贡献,并表明,确切的Shapley值几乎总是很难计算。具体来说,只要至少一个(非冗余)合取词允许长度为3或更长的单词,就很难计算边的贡献。在常规路径查询的情况下(即,没有合取),如果查询只有长度最多为两个的单词,则问题是易处理的;因此,该属性完全表征了问题的易处理性。另一方面,如果我们允许一个近似误差,那么它是直接获得一个有效的计划(FPRAS)的加法近似。然而,乘法近似更难获得。我们建立,在合取正则路径查询的情况下,可以在多项式时间内计算的Shapley值的乘法近似的边缘,当且仅当所有的查询原子是有限的语言(假设非冗余和传统的复杂性限制)。我们还研究了类似的情况下,我们希望确定一个顶点的贡献,而不是一个边缘,并建立类似性质的复杂性结果。

英文摘要:

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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