Capacity Factors of a Point-to-point Network

In this paper, we investigate some properties on capacity factors, which were proposed to investigate the link failure problem from network coding. A capacity factor (CF) of a network is an edge set, deleting which will cause the maximum flow to decrease while deleting any proper subset will not. Generally, a $k$-CF is a minimal (not minimum) edge set which will cause the network maximum flow decrease by $k$. Under point to point acyclic scenario, we characterize all the edges which are contained in some CF, and propose an efficient algorithm to classify. And we show that all edges on some $s$-$t$ path in an acyclic point-to-point acyclic network are contained in some 2-CF. We also study some other properties of CF of point to point network, and a simple relationship with CF in multicast network. On the other hand, some computational hardness results relating to capacity factors are obtained. We prove that deciding whether there is a capacity factor of a cyclic network with size not less a given number is NP-complete, and the time complexity of calculating the capacity rank is lowered bounded by solving the maximal flow. Besides that, we propose the analogous definition of CF on vertices and show it captures edge capacity factors as a special case.