The Ad Types Problem
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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 θ to slot j is v_i·α^θ_j where v_i is an advertiser-specific value and each ad type θ has a discount curve α^(θ)_1>α^(θ)_2> ... > 0 over the slots that is common for ads of type θ. 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 to do VCG pricing in asymptotically the same time, namely O(n^2(k + log n)), and apply reserve prices in 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 θ_i, the next G_ij slots cannot show an ad of type θ_j. We show that the problem is hard to approximate within k^1- ϵ for any ϵ > 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· n^2k + 1) time.
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