Reconstructing a binary matrix under timetabling constraints

Abstract

Our focus is on the problem of reconstructing a $m\times n$ binary matrix M from its vertical and horizontal projections when the following additional constraints have to be satisfied: given an integer $k⩾2$, if $M_{ij}=1$ then $M_{ij-1}=1$ or $M_{ij+1}=1$, and the maximal number of successive 0's on each row of M is not greater than k. We furnish a polynomial time algorithm for reconstructing M by reducing this problem to that of finding a path of given weight in a weighted directed acyclic graph.