Adaptive Submodular Ranking and Routing

TL;DR

This paper introduces a logarithmic approximation algorithm for a broad class of adaptive ranking problems involving submodular functions, with applications in search, active learning, and vehicle routing, achieving near-optimal theoretical guarantees.

Contribution

It develops the first logarithmic approximation algorithm for adaptive submodular ranking and routing, unifying and improving upon previous results in related problems.

Findings

The algorithm achieves a near-optimal approximation ratio of O(log n).

It generalizes and improves bounds for search ranking and active learning problems.

Experimental results demonstrate practical effectiveness in relevant applications.

Abstract

We study a general stochastic ranking problem where an algorithm needs to adaptively select a sequence of elements so as to "cover" a random scenario (drawn from a known distribution) at minimum expected cost. The coverage of each scenario is captured by an individual submodular function, where the scenario is said to be covered when its function value goes above a given threshold. We obtain a logarithmic factor approximation algorithm for this adaptive ranking problem, which is the best possible (unless P=NP). This problem unifies and generalizes many previously studied problems with applications in search ranking and active learning. The approximation ratio of our algorithm either matches or improves the best result known in each of these special cases. Furthermore, we extend our results to an adaptive vehicle routing problem, where costs are determined by an underlying metric. This routing problem is a significant generalization of the previously-studied adaptive traveling salesman and traveling repairman problems. Our approximation ratio nearly matches the best bound known for these special cases. Finally, we present experimental results for some applications of adaptive ranking.