Approximation algorithms for the MAXSPACE advertisement problem

TL;DR

This paper introduces approximation algorithms for the MAXSPACE advertisement scheduling problem and its variants, addressing complex constraints like release dates, deadlines, and values, with guarantees on solution quality.

Contribution

It provides a 1/9-approximation for MAXSPACE-R and a PTAS for MAXSPACE-RDV when the number of slots is constant, advancing solutions for NP-hard scheduling problems.

Findings

A 1/9-approximation algorithm for MAXSPACE-R.

A polynomial-time approximation scheme for MAXSPACE-RDV with constant slots.

MAXSPACE is strongly NP-hard even with two slots.

Abstract

$\newcommand{\cala}{\mathcal{A}}$ In MAXSPACE, given a set of ads $\cala$, one wants to schedule a subset ${\cala'\subseteq\cala}$ into $K$ slots ${B_1, \dots, B_K}$ of size $L$. Each ad ${A_i \in \cala}$ 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 ${A_i \in \cala'}$ appears in exactly $w_i$ slots and at most once per slot. The goal is to find a feasible schedule that maximizes the sum of the space occupied by all slots. We consider a generalization called MAXSPACE-R for which an ad $A_i$ also has a release date $r_i$ and may only appear in a slot $B_j$ if ${j \ge r_i}$. For this variant, we give a $1/9$-approximation algorithm. Furthermore, we consider MAXSPACE-RDV for which an ad $A_i$ also has a deadline $d_i$ (and may only appear in a slot $B_j$ with $r_i \le j \le d_i$), and a value $v_i$ that is the gain of each assigned copy of $A_i$ (which can be unrelated to $s_i$). We present a polynomial-time approximation scheme for this problem when $K$ is bounded by a constant. This is the best factor one can expect since MAXSPACE is strongly NP-hard, even if $K = 2$.