Submodular Optimization with Routing Constraints

Submodular optimization, particularly under cardinality or cost constraints, has received considerable attention, stemming from its breadth of application, ranging from sensor placement to targeted marketing. However, the constraints faced in many real domains are more complex. We investigate an important and very general class of problems of maximizing a submodular function subject to general cost constraints, especially focusing on costs coming from route planning. Canonical problems that motivate our framework include mobile robotic sensing, and door-to-door marketing. We propose a generalized cost-benefit (GCB) greedy algorithm for our problem, and prove bi-criterion approximation guarantees under significantly weaker assumptions than those in related literature. Experimental evaluation on realistic mobile sensing and door-to-door marketing problems, as well as using simulated networks, show that our algorithm achieves significantly higher utility than state-of-the-art alternatives, and has either lower or competitive running time. Introduction There has been much work on submoduar maximization with cardinality constraints (Nemhauser, Wolsey, and Fisher 1978) and additive/modular constraints (Khuller, Moss, and Naor 1999; Sviridenko 2004; Krause and Guestrin 2005; Leskovec et al. 2007). In many applications, however, cost constraints are significantly more complex. For example, in mobile robotic sensing domains, the robot must not only choose where to take measurements, but to plan a route among measurement locations, where costs can reflect battery life. As another example, door-to-door marketing campaigns involve not only the decision about which households to target, but the best route among them, and the constraint reflects the total time the entire effort takes (coming from work schedule constraints). Unlike the typical additive cost constraints, such route planning constraints are themselves NP-Hard to evaluate, necessitating approximation in practice. We tackle the problem of maximizing a submodular function subject to a general cost constraint, c(S) ≤ B, where Copyright c © 2016, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. c(S) is the optimal cost of covering a set S (for example, by a walk through a graph that passes all nodes in S). We propose a generalized cost-benefit greedy algorithm, which adds elements in order of marginal benefit per unit marginal cost. A key challenge is that computing (marginal) cost of adding an element (such as computing the increased cost of a walk when another node is added to a set) is often itself a hard problem. We therefore relax the algorithm to use a polynomial-time approximation algorithm for computing marginal cost. We then show that when the cost function is approximately submodular, we can achieve a bi-criterion approximation guarantee using this modified algorithm, which runs in polynomial time. To our knowledge, this offers the most generally applicable theoretical guarantee in our domain known to date. Our experiments consider two applications: mobile robotic sensors and door-to-door marketing. In the former, we use sensor data on air quality in Beijing, China collected from 36 air quality monitoring stations, with a hypothetical tree-structured routing network among them. The objective in this case is to minimize conditional entropy of unobserved locations, given a Gaussian Process model of joint sensor measurements. In the door-to-door marketing domain, we use rooftop solar adoptions from San Diego county as an example, considering geographic proximity as a social influence network and the actual road network as the routing network. In both these domains, we show that the proposed algorithm significantly outperforms competition, both in terms of achieved utility, and, often, in terms of running time. Remarkably, this is true even in cases where the assumptions in our theoretical guarantees do not meaningfully hold. In summary, this paper makes the following contributions: 1. a formulation of submodular maximization under general cost constraints (routing constraints are of particular interest); 2. a novel polynomial-time generalized cost-benefit algorithm with provable approximation guarantees; 3. an application of our algorithm to two motiving realworld optimization problems, mobile robotic sensing and door-to-door marketing, illustrating that our algorithm significantly outperforms state of the art.