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In this paper we consider the problem of allocating personal TV advertisements to viewers. The problem's input consists of ad requests and viewers. Each ad is associated with a length, a payment, a requested number of viewers, a requested number of allocations per viewer and a target population profile. Each viewer is associated with a profile and an estimated viewing capacity which is uncertain. The goal is to maximize the revenue obtained from the allocation of ads to viewers for multiple periods while satisfying the ad constraints. First, we present the integer programming (IP) models of the problem and several heuristics for the deterministic version of the problem where the viewers' viewing capacities are known in advance. We compare the performances of the proposed algorithms to those of the state-of-the-art IP solver. Later, we discuss the multi-period uncertain problem and, based on the best heuristic for the deterministic version, present heuristics for low and high uncertainty. Through computational experiments, we evaluate our heuristics. For the deterministic version, our best heuristic attains 98 % of the possible revenue and for the multi-period uncertain version our heuristics performances are very high, even in cases of high uncertainty, compared to the revenue obtained by the deterministic version.

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We study a variant of the generalized assignment problem ( gap ), which we label all-or-nothing gap ( agap ). We are given a set of items, partitioned into n groups, and a set of m bins. Each item ℓ has size s ℓ > 0, and utility a ℓ j ⩾ 0 if packed in bin j . Each bin can accommodate at most one item from each group; the total size of the items in a bin cannot exceed its capacity. A group of items is satisfied if all of its items are packed. The goal is to find a feasible packing of a subset of the items in the bins such that the total utility from satisfied groups is maximized. We motivate the study of agap by pointing out a central application in scheduling advertising campaigns. Our main result is an O (1)-approximation algorithm for agap instances arising in practice, in which each group consists of at most m /2 items. Our algorithm uses a novel reduction of agap to maximizing submodular function subject to a matroid constraint. For agap instances with a fixed number of bins, we develop a randomized polynomial time approximation scheme (PTAS) , relying on a nontrivial LP relaxation of the problem. We present a (3 + ε)-approximation as well as PTASs for other special cases of agap , where the utility of any item does not depend on the bin in which it is packed. Finally, we derive hardness results for the different variants of agap studied in this paper.

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