Submodular Function Maximization on the Bounded Integer Lattice

Abstract

The problem of maximizing non-negative submodular functions has been studied extensively in the last few years. However, most papers consider submodular set functions. Recently, several advances have been made for submodular functions on the integer lattice. As a direct generalization of submodular set functions, a function f :for all x, y ∈ {0, . . . , C} n where ∧ and ∨ denote element-wise minimum and maximum. The objective is finding a vector x maximizing f (x). In this paper, we present a deterministic −1)) δ −1 -approximation for every δ > 0 under the assumption that 3 − SAT cannot be solved in time 2 n 3/4+ǫ .

Robust Budget Allocation Via Continuous Submodular Functions

“…Similarly, results for submodular maximization extend to integer lattices, e.g. [Gottschalk & Peis, 2015]. Stronger results are possible if the submodular function also satisfies diminishing returns: for all x ≤ y (coordinate-wise) and i such that y + e i ∈ X , it holds that f (…”

Subspace Selection via DR-Submodular Maximization on Lattices

“…The same definition is introduced by Gottshalk and Peis [11] for distributive lattices. However, this is too strong for our purpose because it cannot capture the principal component analysis; you can check this in Example 1.…”

“…In general, it is NP-hard to obtain a constant factor approximation for a knapsack constrained problem even for a distributive lattice [11]. Therefore, we need additional assumptions on the cost function.…”

“…for all X ≤ Y (component wise inequality) and i ∈ V , where e i is the i-th unit vector. This definition is later extended to distributive lattices [11] and can be extended to general lattices (see Section 3). However, Example 1 above is still crucial, and therefore the objective function of the principal component analysis cannot be an approximated version of a DR-submodular function.…”

Majorisation-minimisation algorithms for minimising the difference between lattice submodular functions

“…In Algorithm 1, note that at every step we are minimising a lattice submodular function, which as shown in [2] to get this to arbitrary precision we have complexity O(( 2GBn ε ) 3 log( 2GBn ε )), for a continuous submodular function defined on [0, B] n with Lipschitz constant G (note that the author minimises a discretised version of the continuous function). For Algorithm 2, we are instead maximising, for which we have an approximation factor of 1/3 [7] and runs in O(kn) calls. For Algorithm 3, we are at each point minimising a modular function, which can be done easily in O(k ′ ) function evaluations, where we have k ′ = i k i , by evaluating each separated function at every point and taking the n minima.…”