A fast and deterministic algorithm for Knapsack-constrained monotone DR-submodular maximization over an integer lattice

We consider a knapsack-constrained maximization problem of a nonnegative monotone DR-submodular function f over a bounded integer lattice [B] in R-+(n), max{f (x) : x is an element of [B] and Sigma(n)(i=1) x(i)c(i) <= 1}, where n is the cardinality of a ground set N and c(.) is a cost function defined on N. Soma and Yoshida [Math. Program., 172 (2018), pp. 539-563] present a (1 - e(-1) - O(epsilon))-approximation algorithm for this problem by combining threshold greedy algorithm with partial element enumeration technique. Although the approximation ratio is almost tight, their algorithm runs in O(n(3)/epsilon(3) log(3) tau[log(3) parallel to B parallel to(infinity) + n/epsilon log parallel to B parallel to(infinity) log 1/epsilon c(min)]) time, where c(min) = min(i) c(i) and tau is the /ratio of the maximum value of f to the minimum nonzero increase in the value of f . Besides, Ene and Nguyen [arXiv:1606.08362, 2016] indirectly give a (1 - e(-1) - O(epsilon))-approximation algorithm with O((1/epsilon)(O(1/epsilon 4))n log parallel to B parallel to(infinity) log(2) (n log parallel to B parallel to(infinity))) time. But their algorithm is random. In this paper, we make full use of the DR-submodularity over a bounded integer lattice, carry forward the greedy idea in the continuous process and provide a simple deterministic rounding method so as to obtain a feasible solution of the original problem without loss of objective value. We present a deterministic algorithm and theoretically reduce its running time to a new record, O((1/epsilon)(O(1 /epsilon 5)) . n log 1/c(min) log parallel to B parallel to(infinity)), with the same approximate ratio.