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

TL;DR

This paper investigates conditions under which $l_p$-norm objective functions in combinatorial optimization can be minimized using graph cuts, extending the applicability beyond the submodular $l_1$-norm case.

Contribution

It provides conditions ensuring $l_p$-norm functions are submodular for all $p \,\geq\, 1$, enabling graph cut minimization for a broader class of problems.

Findings

Identifies conditions for $l_p$-norm submodularity for all $p\geq1$

Expands the class of objective functions solvable by graph cuts

Provides theoretical foundation for $l_p$-norm minimization via graph cuts

Abstract

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