Abstract
In recent years several compressed indexes based on variants of the BurrowsWheeler transformation have been introduced. Some of these are used to index structures far more complex than a single string, as was originally done with the FMindex [Ferragina and Manzini, J. ACM 2005]. As such, there has been an increasing effort to better understand under which conditions such an indexing scheme is possible. This has led to the introduction of Wheeler graphs [Gagie et al., Theor. Comput. Sci., 2017]. Gagie et al. showed that de Bruijn graphs, generalized compressed suffix arrays, and several other BWT related structures can be represented as Wheeler graphs, and that Wheeler graphs can be indexed in a way which is space efficient. Hence, being able to recognize whether a given graph is a Wheeler graph, or being able to approximate a given graph by a Wheeler graph, could have numerous applications in indexing. Here we resolve the open question of whether there exists an efficient algorithm for recognizing if a given graph is a Wheeler graph. We present:
 The problem of recognizing whether a given graph G=(V,E) is a Wheeler graph is NPcomplete for any edge label alphabet of size sigma >= 2, even when G is a DAG. This holds even on a restricted, subset of graphs called dNFA's for d >= 5. This is in contrast to recent results demonstrating the problem can be solved in polynomial time for dNFA's where d <= 2. We also show the recognition problem can be solved in linear time for sigma =1;
 There exists an 2^{e log sigma + O(n + e)} time exact algorithm where n = V and e = E. This algorithm relies on graph isomorphism being computable in strictly subexponential time;
 We define an optimization variant of the problem called Wheeler Graph Violation, abbreviated WGV, where the aim is to remove the minimum number of edges in order to obtain a Wheeler graph. We show WGV is APXhard, even when G is a DAG, implying there exists a constant C >= 1 for which there is no Capproximation algorithm (unless P = NP). Also, conditioned on the Unique Games Conjecture, for all C >= 1, it is NPhard to find a Capproximation;
 We define the Wheeler Subgraph problem, abbreviated WS, where the aim is to find the largest subgraph which is a Wheeler Graph (the dual of the WGV). In contrast to WGV, we prove that the WS problem is in APX for sigma=O(1);
The above findings suggest that most problems under this theme are computationally difficult. However, we identify a class of graphs for which the recognition problem is polynomial time solvable, raising the open question of which parameters determine this problem's difficulty.
BibTeX  Entry
@InProceedings{gibney_et_al:LIPIcs:2019:11172,
author = {Daniel Gibney and Sharma V. Thankachan},
title = {{On the Hardness and Inapproximability of Recognizing Wheeler Graphs}},
booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)},
pages = {51:151:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771245},
ISSN = {18688969},
year = {2019},
volume = {144},
editor = {Michael A. Bender and Ola Svensson and Grzegorz Herman},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/11172},
URN = {urn:nbn:de:0030drops111728},
doi = {10.4230/LIPIcs.ESA.2019.51},
annote = {Keywords: BurrowsWheeler transform, string algorithms, suffix trees, NPcompleteness}
}
Keywords: 

BurrowsWheeler transform, string algorithms, suffix trees, NPcompleteness 
Collection: 

27th Annual European Symposium on Algorithms (ESA 2019) 
Issue Date: 

2019 
Date of publication: 

06.09.2019 