A Polynomial-time Approximation Scheme for the MAXSPACE Advertisement Problem

Abstract

In the MAXSPACE problem, given a set of ads A, one wants to schedule a subset A ′ ⊆ A into K slots B 1 , . . . , B K of size L. Each ad A i ∈ A has a size s i and a frequency w i . A schedule is feasible if the total size of ads in any slot is at most L, and each ad A i ∈ A ′ appears in exactly w i slots. The goal is to find a feasible schedule that maximizes the sum of the space occupied by all slots. We introduce a generalization called MAXSPACE-R in which each ad A i also has a release date r i ≥ 1, and may only appear in a slot B j with j ≥ r i . We also introduce a generalization of MAXSPACE-R called MAXSPACE-RD in which each ad A i also has a deadline d i ≤ K, and may only appear in a slot B j with r i ≤ j ≤ d i . These parameters model situations where a subset of ads corresponds to a commercial campaign with an announcement date that may expire after some defined period. We present a 1/9-approximation algorithm for MAXSPACE-R and a polynomial-time approximation scheme for MAXSPACE-RD when K is bounded by a constant. This is the best factor one can expect, since MAXSPACE is NP-hard, even if K = 2.

Local-Search Based Heuristics for Advertisement Scheduling

“…Amiri and Menon [2] present an integer linear programming to a MAXSPACE variant in which each ad has a set of values for frequency. Da Silva et al [4] present a polynomial-time approximation scheme for MAXSPACE with deadlines and release dates and with a constant number of slots.…”

“…Algorithm LSLF is also a ( 43 − 1 3K/w )-approximation to MIN K|w [5]. Dawande et al [5] present a 2-approximation for MINSPACE using LP Rouding, and Dean and Goemans [6] present a 4 3 -approximation for MINSPACE using Graham's algorithm for schedule [17].…”