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Techniques based on minimal graph cuts have become a standard tool forsolving combinatorial optimization problems arising in image processing andcomputer vision applications. These techniques can be used to minimizeobjective functions written as the sum of a set of unary and pairwise terms,provided that the objective function is submodular. This can be interpreted asminimizing the $l_1$-norm of the vector containing all pairwise and unaryterms. By raising each term to a power $p$, the same technique can also be usedto minimize the $l_p$-norm of the vector. Unfortunately, the submodularity ofan $l_1$-norm objective function does not guarantee the submodularity of thecorresponding $l_p$-norm objective function. The contribution of this paper isto provide useful conditions under which an $l_p$-norm objective function issubmodular for all $p\geq 1$, thereby identifying a large class of $l_p$-normobjective functions that can be minimized via minimal graph cuts.

When can $l_p$-norm objective functions be minimized via graph cuts?