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Minimum Cost Adaptive Submodular Cover

Hessa Al-Thani, Yubing Cui, Viswanath Nagarajan

TL;DR

This work studies minimum-cost adaptive-submodular cover, introducing a greedy policy that achieves a $4\,(1+\ln Q)$-approximation for the base problem and a universal $(p+1)^{p+1}\,(1+\ln Q)^p$-approximation for all $p\ge1$ on the $p^{th}$ moment objective. It develops a novel non-completion-probability analysis to relate greedy and optimal policies, and extends these guarantees to the multi-function setting (MASC). The authors also provide applications to stochastic submodular cover, adaptive viral marketing, and optimal decision trees with uniform priors, supported by empirical results on real data. The results tighten known bounds for adaptive-submodular cover, unify moment-based objectives under a single algorithm, and establish near-optimal constants for a broad class of stochastic coverage problems, with practical impact for sensor placement, learning, and influence maximization tasks.

Abstract

Adaptive submodularity is a fundamental concept in stochastic optimization, with numerous applications such as sensor placement, hypothesis identification and viral marketing. We consider the problem of minimum cost cover of adaptive-submodular functions, and provide a $4(1+\ln Q)$-approximation algorithm, where $Q$ is the goal value. In fact, we consider a significantly more general objective of minimizing the $p^{th}$ moment of the coverage cost, and show that our algorithm simultaneously achieves a $(p+1)^{p+1}\cdot (\ln Q+1)^p$ approximation guarantee for all $p\ge 1$. All our approximation ratios are best possible up to constant factors (assuming $P\ne NP$). Moreover, our results also extend to the setting where one wants to cover {\em multiple} adaptive-submodular functions. Finally, we evaluate the empirical performance of our algorithm on instances of hypothesis identification.

Minimum Cost Adaptive Submodular Cover

TL;DR

This work studies minimum-cost adaptive-submodular cover, introducing a greedy policy that achieves a -approximation for the base problem and a universal -approximation for all on the moment objective. It develops a novel non-completion-probability analysis to relate greedy and optimal policies, and extends these guarantees to the multi-function setting (MASC). The authors also provide applications to stochastic submodular cover, adaptive viral marketing, and optimal decision trees with uniform priors, supported by empirical results on real data. The results tighten known bounds for adaptive-submodular cover, unify moment-based objectives under a single algorithm, and establish near-optimal constants for a broad class of stochastic coverage problems, with practical impact for sensor placement, learning, and influence maximization tasks.

Abstract

Adaptive submodularity is a fundamental concept in stochastic optimization, with numerous applications such as sensor placement, hypothesis identification and viral marketing. We consider the problem of minimum cost cover of adaptive-submodular functions, and provide a -approximation algorithm, where is the goal value. In fact, we consider a significantly more general objective of minimizing the moment of the coverage cost, and show that our algorithm simultaneously achieves a approximation guarantee for all . All our approximation ratios are best possible up to constant factors (assuming ). Moreover, our results also extend to the setting where one wants to cover {\em multiple} adaptive-submodular functions. Finally, we evaluate the empirical performance of our algorithm on instances of hypothesis identification.
Paper Structure (31 sections, 10 theorems, 59 equations, 3 figures, 1 table, 2 algorithms)

This paper contains 31 sections, 10 theorems, 59 equations, 3 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. Random items.
  4. Utility function.
  5. Min-cost adaptive-submodular cover (${\sf ASC}$).
  6. Remark:
  7. Adaptive Greedy Policy
  8. Remark:
  9. Results and Techniques
  10. Related Work
  11. Analyzing the Greedy Policy
  12. Non-completion Probabilities and the Moment Objective
  13. Using the Greedy Criterion
  14. Truncated optimal policy
  15. Conditioning on partial realizations in greedy
  16. ...and 16 more sections

Key Result

Theorem 1.5

Consider any instance of minimum cost adaptive-submodular cover, where the utility function $f:2^{E\times\Omega}\to\mathbb R_{\ge 0}$ is monotone, coverable and adaptive-submodular w.r.t. the probability distribution $p(\cdot)$. Suppose that there is some value $\eta>0$ such that $f(\psi)>Q-\eta$ im where $\sigma$ denotes the optimal policy.

Figures (3)

  • Figure 1: Graph of a simple $o(\cdot)$ function.
  • Figure 2: Graph of a simple $score(t,\psi)$ for illustration. $e_1, e_2,...$ are greedy selections and $\psi_i$ is the partial realization just before selecting $e_i$.
  • Figure 3: Example of $g(x)$, where $k=5,Q=10, \eta=1$ and $f_1=0, f_2=2, f_3=3.5, f_4=7, f_5=8.5$.

Theorems & Definitions (25)

  • Definition 1.1: Monotonicity
  • Definition 1.2: Coverable
  • Definition 1.3: Marginal benefit
  • Definition 1.4: Adaptive submodularity
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1: Non-completion probabilities
  • Lemma 2.2
  • proof
  • Definition 2.3: Greedy score
  • ...and 15 more