The Ad Types Problem

Riccardo Colini-Baldeschi, Julián Mestre, Okke Schrijvers, Christopher A. Wilkens

Abstract: In this paper we introduce the Ad Types Problem, a generalization of the traditional positional auction model for ad allocation that better captures some of the challenges that arise when ads of different types need to be interspersed within a user feed of organic content. The Ad Types problem (without gap rules) is a special case of the assignment problem in which there are k types of nodes on one side (the ads), and an ordered set of nodes on the other side (the slots). The edge weight of an ad i of type $$\theta $$ to slot j is $$v_i\cdot \alpha ^{\theta }_j$$ where $$v_i$$ is an advertiser-specific value and each ad type $$\theta $$ has a discount curve $$\alpha ^{(\theta )}_{1} \ge \alpha ^{(\theta )}_{2} \ge \ldots \ge 0$$ over the slots that is common for ads of type $$\theta $$ . We present two contributions for this problem: 1) we give an algorithm that finds the maximum weight matching that runs in $$O(n^2(k + \log n))$$ time for n slots and n ads of each type—cf. $$O(kn^3)$$ when using the Hungarian algorithm—, and 2) we show how to apply reserve prices in total time $$O(n^3(k + \log n))$$ . The Ad Types Problem (with gap rules) includes a matrix G such that after we show an ad of type $$\theta _i$$ , the next $$G_{ij}$$ slots cannot show an ad of type $$\theta _j$$ . We show that the problem is hard to approximate within $$k^{1- \epsilon }$$ for any $$\epsilon > 0$$ (even without discount curves) by reduction from Maximum Independent Set. On the positive side, we show a Dynamic Program formulation that solves the problem (including discount curves) optimally and runs in $$O(k\cdot n^{2k + 1})$$ time.