关于Wheeler图识别的困难性和不可逼近性
On the Hardness and Inapproximability of Recognizing Wheeler Graphs
摘要：
近年来，几个压缩指数的基础上的变体的Burrows-Wheeler变换已被引入。这些索引结构中的一些比单个字符串复杂得多，就像最初使用FM索引所做的那样[Ferragina and Manzini，J. ACM 2005]。因此，一直在努力更好地理解在哪些条件下这种索引方案是可能的。这导致了Wheeler图的引入[Gagie it et al.，Theor. Comput.科学，2017年]。Wheeler图是满足两个简单公理的带边标号的有向图。Wheeler图可以以一种空间有效并允许快速遍历的方式进行索引。Gagie等人证明了de Bruijn图、广义压缩后缀数组和其他几个BWT相关结构可以表示为Wheeler图。在这里，我们回答的开放性问题，是否存在一个有效的算法来识别，如果一个图是惠勒图。我们证明：（i）识别一个图是否是惠勒图是NP-完全的大小为$σ\geq 2$的任何边标签字母表，即使是DAG。它可以解决线性时间为$σ=1$ ;（ii）一个优化的变体称为惠勒图违规（WGV），其目的是删除最小数量的边缘需要获得一个惠勒图是APX硬，即使是DAG。因此，除非P = NP，否则存在常数$C > 1$，使得不存在$C$ -近似算法。我们证明了在唯一博弈猜想的条件下，对于每个常数$C \geq 1$，找到WGV的$C$ -近似是NP-困难的;（iii）寻找最大Wheeler子图的Wheeler子图问题（WS）在$σ=O(1)$的APX中;（iv）对于上述问题，存在有效的指数时间精确算法，依赖于在严格的次指数时间内计算图同构;（v）识别问题是多项式时间可解的一类图。
英文摘要：
In recent years several compressed indexes based on variants of the Burrows-Wheeler transformation have been introduced. Some of these index structures far more complex than a single string, as was originally done with the FM-index [Ferragina and Manzini, J. ACM 2005]. As such, there has been an effort to better understand under which conditions such an indexing scheme is possible. This led to the introduction of Wheeler graphs [Gagie it et al., Theor. Comput. Sci., 2017]. A Wheeler graph is a directed graph with edge labels which satisfies two simple axioms. Wheeler graphs can be indexed in a way which is space efficient and allows for fast traversal. Gagie et al. showed that de Bruijn graphs, generalized compressed suffix arrays, and several other BWT related structures can be represented as Wheeler graphs. Here we answer the open question of whether or not there exists an efficient algorithm for recognizing if a graph is a Wheeler graph. We demonstrate:(i) Recognizing if a graph is a Wheeler graph is NP-complete for any edge label alphabet of size $σ\geq 2$, even for DAGs. It can be solved in linear time for $σ=1$; (ii) An optimization variant called Wheeler Graph Violation (WGV) which aims to remove the minimum number of edges needed to obtain a Wheeler graph is APX-hard, even for DAGs. Hence, unless P = NP, there exists constant $C > 1$ such that there is no $C$-approximation algorithm. We show conditioned on the Unique Games Conjecture, for every constant $C \geq 1$, it is NP-hard to find a $C$-approximation to WGV; (iii) The Wheeler Subgraph problem (WS) which aims to find the largest Wheeler subgraph is in APX for $σ=O(1)$; (iv) For the above problems there exist efficient exponential time exact algorithms, relying on graph isomorphism being computed in strictly sub-exponential time; (v) A class of graphs where the recognition problem is polynomial time solvable.
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