We show that the problem of whether a query is equivalent to a query of tree-width $k$ is decidable, for the class of Unions of Conjunctive Regular Path Queries with two-way navigation (UC2RPQs). A previous result by Barcel\'o, Romero, and Vardi [SIAM Journal on Computing, 2016] has shown decidability for the case $k=1$, and here we extend this result showing that decidability in fact holds for any arbitrary $k\geq 1$. The algorithm is in 2ExpSpace, but for the restricted but practically relevant case where all regular expressions of the query are of the form $a^*$ or $(a_1 + \dotsb + a_n)$ we show that the complexity of the problem drops to $\Pi^P_2$. We also investigate the related problem of approximating a UC2RPQ by queries of small tree-width. We exhibit an algorithm which, for any fixed number $k$, builds the maximal under-approximation of tree-width $k$ of a UC2RPQ. The maximal under-approximation of tree-width $k$ of a query $q$ is a query $q'$ of tree-width $k$ which is contained in $q$ in a maximal and unique way, that is, such that for every query $q''$ of tree-width $k$, if $q''$ is contained in $q$ then $q''$ is also contained in $q'$. Our approach is shown to be robust, in the sense that it allows also to test equivalence with queries of a given path-width, it also covers the previously known result for $k=1$, and it allows to test for equivalence of whether a (one-way) UCRPQ is equivalent to a UCRPQ of a given tree-width (or path-width).

翻译：我们证明了对于带双向导航的合取正则路径查询的并集（UC2RPQs），判断一个查询是否等价于树宽为$k$的查询是可判定的。Barceló、Romero和Vardi的先前结果[SIAM Journal on Computing, 2016]已证明$k=1$情况的可判定性，而本文将该结果推广到任意$k\geq 1$。该算法处于2ExpSpace复杂度，但对于所有正则表达式形如$a^*$或$(a_1 + \dotsb + a_n)$的受限但实际相关情形，问题复杂度可降至$\Pi^P_2$。我们还研究了用小树宽查询近似UC2RPQ的相关问题。针对任意固定数$k$，我们提出了一种算法，该算法能构建UC2RPQ的最大树宽$k$下逼近。查询$q$的最大树宽$k$下逼近是树宽为$k$的查询$q'$，它以最大且唯一的方式包含于$q$中，即对于每个树宽为$k$的查询$q''$，若$q''$包含于$q$则$q''$也包含于$q'$。我们的方法具有鲁棒性：既能检验与给定路径宽查询的等价性，又覆盖了$k=1$的已知结果，还可检验（单向）UCRPQ是否等价于给定树宽（或路径宽）的UCRPQ。