A fast and deterministic algorithm for Knapsack-constrained monotone DR-submodular maximization over an integer lattice - Gong, Suning; Nong, Qingqin; Bao, Shuyu; Fang, Qizhi; Du, Ding-Zhu (Journal of global optimization, 2023)
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 ∈ [ B ] and ∑ i = 1 n 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 ( ϵ ) ) -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 ϵ 3 log 3 τ [ log 3 B ∞ + n ϵ log B ∞ log 1 ϵ c min ] ) time, where c min = min i c ( i ) and τ is the ratio of the maximum value of f to the minimum nonzero increase in the value of f . Besides, Ene and Nguy e ˇ ~ n [ arXiv:1606.08362 , 2016] indirectly give a ( 1 - e - 1 - O ( ϵ ) ) -approximation algorithm with O ( ( 1 ϵ ) O ( 1 / ϵ 4 ) n log ‖ B ‖ ∞ log 2 ( n log ‖ B ‖ ∞ ) ) 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 ϵ ) O ( 1 / ϵ 5 ) · n log 1 c min log ‖ B ‖ ∞ ) , with the same approximate ratio.