Approximation Algorithms for Matroidal Prerequisite Systems

Abstract

Optimal selections in a decision process are often constrained by prerequisites. However, such prerequisites can encode functional rather than literal dependencies, so a required dependency may be supplied by one or several interacting alternatives. We introduce matroidal prerequisite systems (MPS), a constraint structure where a poset specifies prerequisites while a matroid determines when those prerequisites have been satisfied by its span. This creates an order-sensitive notion of feasibility over words, where feasible words are associated with independent sets, while dependencies may be fulfilled through substitutable functionality. Our main contribution is approximation algorithms for additive maximization and submodular maximization over the feasible words of an MPS. The guarantees are determined by two structural parameters: the maximum matroid rank $\Delta$ of a principal ideal in the poset and the maximum matroid connectivity $\lambda_\mathrm{max}$. These measure the distance an MPS is from encoding a matroid or a poset antimatroid, respectively, both of which are generalized by an MPS. For additive maximization, we obtain efficient deterministic $\Delta$- and $(1+\lambda_\mathrm{max})$-approximation algorithms. By extending these techniques, we obtain efficient deterministic $(2+\lambda_\mathrm{max})$-approximation and randomized $(\Delta^2\cdot(1 - 1/e - \delta)^{-1})$-approximation algorithms for all $\delta >0$ for submodular maximization. The algorithm design and analysis use the theory of polymatroid greedoids, via cryptomorphism we prove between an MPS and a strong polymatroid greedoid. Finally, an approximation-preserving reduction from densest $k$-subgraph shows it is not possible to efficiently compute a $\min\{\Delta,\lambda_\mathrm{max}\}^{o(1)}$-approximation to additive maximization over the feasible words of an MPS under the Gap Exponential Time Hypothesis.