Interactive Submodular Set Cover

We introduce a natural generalization of submodular set cover and exact active learning with a finite hypothesis class (query learning). We call this new problem interactive submodular set cover. Applications include advertising in social networks with hidden information. We give an approximation guarantee for a novel greedy algorithm and give a hardness of approximation result which matches up to constant factors. We also discuss negative results for simpler approaches and present encouraging early experimental results.

💡 Research Summary

The paper introduces a novel optimization problem called Interactive Submodular Set Cover (ISSC), which unifies two well‑studied domains: submodular set cover and exact active learning (query learning) over a finite hypothesis class. In classic submodular set cover, one seeks a minimum‑cost subset of elements whose associated submodular function reaches a prescribed threshold. In exact active learning, the goal is to ask a limited number of queries that uniquely identify an unknown hypothesis from a known hypothesis set. ISSC blends these objectives: at each step the algorithm may select an element (e.g., a user to target with an advertisement) that incurs a cost, updates the set of hypotheses still consistent with the observations, and simultaneously accrues submodular “coverage” (such as influence spread or information gain). The problem is motivated by applications where hidden information must be uncovered while a resource‑limited campaign proceeds, for example advertising in social networks with unknown user preferences.

Formal Model

• Hypothesis space (H) with prior distribution (\pi).
• Ground set (E) of selectable actions (queries, ads, sensor placements).
• For each hypothesis (h\in H) a monotone submodular cost function (f_h:2^{E}\rightarrow\mathbb{R}_{\ge0}) and a per‑action cost (c(e)).
• The algorithm proceeds adaptively: after a sequence (S_t) of selected actions it observes the induced information set (I_t = {h\in H\mid S_t\text{ is consistent with }h}).
• Goal: find an adaptive policy that, with probability 1, reduces the consistent hypothesis set to a singleton and drives every (f_h(S)) above a given threshold (\tau_h), while minimizing the expected total cost.

Greedy Algorithm

The authors propose a greedy policy based on expected marginal gain per unit cost. At any state (I_t) the algorithm computes for each candidate action (e): \