The complexity of matrix completion

Nicholas J. A. Harvey, David R. Karger, Sergey Yekhanin
2006 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm - SODA '06  
Given a matrix whose entries are a mixture of numeric values and symbolic variables, the matrix completion problem is to assign values to the variables so as to maximize the resulting matrix rank. This problem has deep connections to computational complexity and numerous important algorithmic applications. Determining the complexity of this problem is a fundamental open question in computational complexity. Under different settings of parameters, the problem is variously in P, in RP, or
more » ... We shed new light on this landscape by demonstrating a new region of NP-hard scenarios. As a special case, we obtain the first known hardness result for matrices in which each variable appears only twice. Another particular scenario that we consider is the simultaneous matrix completion problem, where one must simultaneously maximize the rank for several matrices that share variables. This problem has important applications in the field of network coding. Recent work has given a simple, greedy, deterministic algorithm for this problem, assuming that the algorithm works over a sufficiently large field. We show an exact threshold for the field size required to find a simultaneous completion efficiently. This result implies that, surprisingly, the simple greedy algorithm is optimal: finding a simultaneous completion over any smaller field is NPhard. * Supported by a Natural Sciences and Engineering Research Council of Canada PGS Scholarship.
doi:10.1145/1109557.1109679 fatcat:oqhcbk3vjnd4xabxqbnyex6br4