Posimodular Function Optimization

Abstract

A function Open image in new window on a finite set V is posimodular if \(f(X)+f(Y) \ge f(X\setminus Y)+f(Y\setminus X)\), for all \(X,Y\subseteq V\). Posimodular functions often arise in combinatorial optimization such as undirected cut functions. We consider the problem of finding a nonempty subset X minimizing f(X), when the posimodular function f is given by oracle access.

We show that posimodular function minimization requires exponential time, contrasting with the polynomial solvability of submodular function minimization that forms another generalization of cut functions. On the other hand, the problem is fixed-parameter tractable in terms of the size of the image (or range) of f.

In more detail, we show that \(\varOmega (2^{0.3219n} T_f)\) time is necessary and \(O(2^{0.92n}T_f)\) sufficient, where \(T_f\) denotes the time for one function evaluation. When the image of f is \(D=\{0,1,\ldots ,d\}\), \(O(2^{1.271d}nT_f)\) time is sufficient and \(\varOmega (2^{0.1609d}T_f)\) necessary. We can also generate all sets minimizing f in time \(2^{O(d)} n^2 T_f\).

Finally, we also consider the problem of maximizing a given posimodular function, showing that it requires at least \(2^{n-1}T_f\) time in general, while it has time complexity \(\varTheta (n^{d-1}T_f)\) when \(D=\{0,1,\ldots , d\}\) is the image of f, for integer d.