Abstract
The incidence matrix of a graph is a fundamental object naturally appearing in many applications, involving graphs such as social networks, communication networks, or transportation networks. Often, the data collected about the incidence relations can have some slight noise. In this paper, we initiate the study of the computational complexity of recovering incidence matrices of graphs from a binary matrix: given a binary matrix M which can be written as the superposition of two binary matrices L and S, where S is the incidence matrix of a graph from a specified graph class, and L is a matrix (i) of small rank or, (ii) of small (Hamming) weight. Further, identify all those graphs whose incidence matrices form part of such a superposition. Here, L represents the noise in the input matrix M. Another motivation for this problem comes from the Matroid Minors project of Geelen, Gerards and Whittle, where perturbed graphic and cographic matroids play a prominent role. There, it is expected that a perturbed binary matroid (or its dual) is presented as L+S where L is a low rank matrix and S is the incidence matrix of a graph. Here, we address the complexity of constructing such a decomposition.
When L is of small rank, we show that the problem is NPcomplete, but it can be decided in time (mn)^O(r), where m,n are dimensions of M and r is an upperbound on the rank of L. When L is of small weight, then the problem is solvable in polynomial time (mn)^O(1). Furthermore, in many applications it is desirable to have the list of all possible solutions for further analysis. We show that our algorithms naturally extend to enumeration algorithms for the above two problems with delay (mn)^O(r) and (mn)^O(1), respectively, between consecutive outputs.
BibTeX  Entry
@InProceedings{fomin_et_al:LIPIcs:2020:12916,
author = {Fedor V. Fomin and Petr Golovach and Pranabendu Misra and M. S. Ramanujan},
title = {{On the Complexity of Recovering Incidence Matrices}},
booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)},
pages = {50:150:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771627},
ISSN = {18688969},
year = {2020},
volume = {173},
editor = {Fabrizio Grandoni and Grzegorz Herman and Peter Sanders},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12916},
URN = {urn:nbn:de:0030drops129164},
doi = {10.4230/LIPIcs.ESA.2020.50},
annote = {Keywords: Graph Incidence Matrix, Matrix Recovery, Enumeration Algorithm}
}
Keywords: 

Graph Incidence Matrix, Matrix Recovery, Enumeration Algorithm 
Collection: 

28th Annual European Symposium on Algorithms (ESA 2020) 
Issue Date: 

2020 
Date of publication: 

26.08.2020 