QPBO produces a solution where each variable assumes one of three possible values,: ''true'', ''false'', and ''undefined'', noted in the following as 1, 0, and <math>\emptyset</math> respectively. The solution satisfieshas the two following two properties:.
* ''Partial optimality'': if <math>f</math> is submodular, then QPBO produces a global minimum exactly, equivalentlyequivalent to [[graph cut optimization|graph cut]], and all variables have a non-undefined value.; Ifif submodularity is not satisfied, the result will be a partial solution <math>\mathbf{x}</math> where a subset <math>\hat{V} \subseteq V</math> of the variables have a non-undefined value. SuchA partial solution is always part of a global solution, i.e. there existexists a global minimum point <math>\mathbf{x^*}</math> for <math>f</math> such that <math>x_i = x_i^*</math> for each <math>i \in \hat{V}</math>.
* ''persistencePersistence'': given a solution <math>\mathbf{x}</math> generated by QPBO and an arbitrary assignment of values <math>\mathbf{y}</math> to the variables, if a new solution <math>\hat{\mathbf{y}}</math> is constructed by replacing <math>y_i</math> with <math>x_i</math> for each <math>i \in \hat{V}</math>, then <math>f(\hat{\mathbf{y}}) \le f(\mathbf{y})</math>.<ref name="review" />
