A note on sparse solutions of sparse linear systems

Abstract A vector with at most k nonzeros is called k -sparse. Given a linear system A x = b (the rows of A are r -sparse), it has been shown by Damaschke [1] that enumerating nonnegative k -sparse solutions is fixed-parameter tractable. They present an algorithm reaching an O ⁎ ( r 2 k ) time bound. Since this problem is closely related to the hitting set problem, they raise the question whether it is possible to improve this time bound to O ⁎ ( r k ) . This paper investigates this problem and discovers that a refined analysis of a modified version of their algorithm leads to a smaller time bound O ⁎ ( ( 4 r ) k ) .