A 1/2-approximation algorithm for maximizing a non-monotone weak-submodular function on a bounded integer lattice

Maximizing non-monotone submodular functions is one of the most important problems in submodular optimization. Let B=(B1,B2, horizontal ellipsis ,Bn)is an element of Z+n be an integer vector and [B]={(x1,MIDLINE HORIZONTAL ELLIPSIS,xn)is an element of Z+n:0 <= xk <= Bk,for all 1 <= k <= n} be the set of all non-negative integer vectors not greater than B A function f:[B]-> Ris said to be weak-submodular if f(x+delta 1k)-f(x)>= f(y+delta 1k)-f(y) for any k is an element of{1,MIDLINE HORIZONTAL ELLIPSIS,n}any pair of x,y is an element of[B]such that x <= yand xk=ykand any delta is an element of Z+satisfying y+delta 1k is an element of[B] Here 1k is the vector with the kth component equal to 1 and each of the others equals to 0. In this paper we consider the problem of maximizing a non-monotone and non-negative weak-submodular function on the bounded integer lattice without any constraint. We present an randomized algorithm with an approximation guarantee 12 for the problem.