Proof of the NP-hardness of Topology Cuts problem

s.t. T = Tinit with xp ∈ {0, 1}, p ∈ V. Here V is the regular image grid domain, and E denotes the 4or 8neighborhood in the image. The terms of the above energy function have the form D(xp) = E1 p(1− xp) + E0 pxp and Vpq(xp, xq) = wpq((1 − xp)xq + (1 − xq)xp) with wpq ≥ 0. In the image domain, we consider the case that all the 0’s (foreground) are 4-connected and all the 1’s (background) are 8-connected. Here T denotes the topology of the 0/1 labeled image as defined in [1]. Tinit is the initial topology information. The above energy minimization problem can also be implemented on a graph G with two extra nodes – source and sink (Fig 1(a)). A solution to the energy function (1), i.e., an 0/1 assignment to each variable xi, corresponds to a cut on the graph (Fig 1(b)). Minimizing the energy function (1) is equivalent to finding a minimal cut on the graph G with given topology constraint. Note that under the given topology constraint, a minimal cut does not necessarily correspond to a max-flow on the graph.